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when it is extended to the whole surface of the sphere, and in the particular circumstance of r = p, or r—p=0. We must begin with transforming the formula to be integrated. The arcs 0 and 0′ are the two sides of a triangle formed on the surface of a sphere; the angle contained by those sides is '-w; and the third side of the same triangle is no other than the arc whose cosine has been denoted by y: let o denote the angle opposite to the side 6' whose cosine is '; then if we suppose ' and 'to vary, it has already been proved that the correspondent fluxion of the surface of the sphere will be = p. du'. d'; but if we make y and o vary, the same fluxion will be = p2. dy. do: therefore

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and as this is true for every element of the spherical surface, the fluents will likewise be equal when they are extended to the whole surface of the sphere. To complete the transformation we must next convert d' into a function ofy and ; after which the integration with regard to will be independent of the denominator in which y only is contained. Suppose v to be actually transformed as here mentioned, then

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the sign of integration in the numerator being understood to affect the variable only. P

For the greater simplicity we shall first consider the case when u' is a rational and integral function of 'only without ', as is the case in spheroids of revolution. Suppose then = F(u'): and by spherical trigonometry,

μ'= μy + √ 1 — μ3. √1 — ya . cos. q ; therefore by TAYLOR's theorem,

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and by substituting for the powers of cos. their values in the multiple arcs, we shall have,

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Ƒ‹°) + (1 — — μ3)3 . ( 1 — y3)3 . Ƒ(1). cos. ¢

+ (1 —μ3)3 . (1 — 2o)3. r(2). cos. 24 + &c.

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the general term being ( 1 — μ2 ) . ( 1 — y2)a × r‹3. cos. ¿q,

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where r represents a rational and integral function of 2. Now if we multiply by do, and then integrate between the limits o̟ =0 and 4 = 2′′, we shall get sv'do = 2+ . F(°); because the integrals of all the terms which contain the cosines of the multiple arcs are evanescent at both the limits. Therefore, by substitution,

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In order to execute the remaining integration I remark that

ƒ = {r— 2rp . y +p2 } 3, and dy = -d: therefore by conti


dr. F (μy)

* By the notation

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it is to be understood that in taking the fluxions,

(y) is to be considered as one simple quantity; the same as if it were represented by a single letter.

D &

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This fluent, which, it is to be observed, increases as y increases, is to be taken between the limits = 1 and y = 1: at the first limit y=-1, every term of the fluent is evanescent when rpo: at the second limit, y = 1 and ƒ = r — p, every term is likewise evanescent except the first, which is (r—p)i—1_r(0) (r—p)?—1 × ¡—1

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= 27. for all values of r, and even

when r-po: therefore

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observing that we must make y = 1 in the function (). Now the suppositions y 1 and y=1, correspond to μ'μ μ'— — μ and μμ: and therefore if we put u to denote the same function of μ that 'does of '; that is, if v represent what




becomes when μ'μ; then it will follow, from the nature of μ'= =μ; the transformed value of u', that v= =r(°) when =1, because

all the other terms are equal to nothing for this value of y: therefore finally

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We shall now pass on to the general case when is a


rational and integral function of μ', V1—μ22. cos. ',

. √ 1


sin. w'. Let x, y, z stand for μ, ✔― μ. cos. ', — μ sin. '; and x', y', z' for the analogous magnitudes y,

. cos. 4, √1 — y3. sin. : the first set of quantities are three rectangular co-ordinates of a point in the surface of the sphere whose radius is unit, drawn to the planes of three great circles two of which intersect in the origin of the arcs whose cosines are μ and μ; and the second set are the three rectangular co-ordinates of the same point as before referred to three other planes two of which pass through the origin of the arc whose cosine is y: therefore, in order to obtain the relation of these two sets of quantities we have only to apply the method for transforming the co-ordinates: in this manner we shall readily obtain,

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— z'. sin.

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y = x'. √1 — μ3. cos. —y'. μ. cos. ☎ —


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z = x′' . √ 1 — μ3 . sin. w — y' . μ . sin. w + z' . cos. w. Because' is a rational and integral function of x, y, z; by substituting the values of these quantities just investigated, it will be converted into a like function of x', y', z', that is, of y, √1-y3. cos. 4, ✔y. sin. q: and farther, if the several powers and products of cos. o and sin. o be exterminated by means of the equivalent expressions in the sines and cosines of the multiple arcs, the expression u', after all the terms are properly arranged, will assume the following form, viz. v=s(0) ¿'= Ƒ(°) + ( 1 − μ3) 3 . ( 1 — y2)2. Ƒ(1). cos. 4 + ( 1 −μ2 )3 . (1 — 23)3. r(2). cos. 20 + &c.

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+ (1 − μ3)* . ( 1 − 2o)2. ▲(1). sin. ¢ + ( 1 — μ• )2. (1 — z• )3 A(2). sin. 29+ &c.

the general term being

(1—μ3)‡. (1—22)‡. г. cos. ¿q + (1—μ3)*. (1—2°)‡. A@


. sin. iq,

(i) A

where r and represent rational and integral functions. of y. Now if we multiply by do and then integrate from =0 to 4 = 27, we shall obtain as beforeƒv'do = 2′′ × г(°);

because the integrals of all the terms multiplied by the cosines and sines of the multiple arcs are of the same magnitude at both the limits. Therefore, by following exactly the same procedure as before, we shall arrive at this equation, viz.

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in which the function () is to be valued on the supposition that y=1. But the suppositions y=-1, p=0, correspond to μ'— — μ, Ɗ'=; and the suppositions y=1, 4 = 2′′, correspond to u': - fh, and ☎'= ☎ + 2π: therefore if u denote what u becomes when uμ and ʊ'=□ + 2′′; that is if v

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be the same function of , V1-μ2. cos.,


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that v' is of μ', √1— μ22. cos. ', √1—2. sin. '; it is r() when

plain, from the transformed value of u', that u =

y= 1. Therefore, we shall have

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4. The investigation just gone through shows how necessary it is to retain all the terms' we have done in the equation (C), and at the same time it proves that the terms thrown out in finding that equation were justly rejected. It completely

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