530 MR. IVORY ON A MASS OF HOMOGENEOUS FLUID AT LIBERTY. 



For the sake of brevity put cos 6 = a, sin 6 cos ra- = ^1 — a^ cos ts = h, sin 6 sin w 

 = ^1 — «2 gju rjr = c ; and from the last expressions we readily deduce 



a! = cos & = a.y-\->^\.— a^.^l— y^ cos a 



b' = sin & cos ts-' = & . y — a cos cr . ^y 1 — y^ cos a ■\-^\\\w ^\ — y^ sin (r 

 c' = sin ^' sin ?zr' = c . y — a sin C7 . ^1 — y^ cos <r — cos ?:r . ^^y 1 — y^ sin c. 

 If these values be substituted in 



and the several powers be expanded and reduced to terms containing the sines and 

 cosines of the multiples of the arc <r, the result will be of this form : 



r^^^ + (1 - y2)^. r^'^ cos ^ + (1 - y2)" . r^'^ cos 2 a +,&c. 

 + (1 ~ y2)" A^'^ sin 0- + (1 - y^f A^^^ sin 2 (r +, &c., 



the expressions F^ and A^'^ being integral functions of y ; and it is to be observed 

 that the index of the highest power of y in P° cannot exceed m ■\- m! -{- m". If we 

 now multiply by dff and integrate between the limits c = and <r = 2 t, we shall get 



J a b c . a c = 2 T X i . 

 Wherefore, attending to the expression of C^*\ we have 



J-dydaC a' b' c' =^. i . 2 . 3 . . . z 7- ^^ ^ ~7/ ' 

 Now it is easy to prove that 



when n is less than i, the integral being taken between the limits y = + i and 

 y = — 1 : and since the highest power of y in P° does not exceed m + m' + m", it 

 must be less than i ; and hence it follows that the integral under consideration is 

 equal to zero. 



