80 MR. C. J. HARGREAVE ON THE CALCULATION OF ATTRACTIONS, 



and by repeating a similar process, we obtain 



d^q , dq ^^ X-Y , c-6 



dTdX + ^TX ^ *^'^~"2~ + ^ 4.„«2X- Y - ^• 



4 CDS'* 



2 

 By observing the assumptions here made, and the results obtained, we find that in 



the first assumption \-Ty = ^)> the coeflScient of P^ tan — ^ — i^ ^» ^^ ^^ second, 



X — Y 



that of V tan — ^ — = 1 J ^^^d so on, in the order of the natural numbers ; and in the 



, ., rf - ,flf^« — 2„a — 6 „ a—nin — X) 

 results, the numerical coefficients are 1, -^\ 2, — ^ — ; 3, ^ . . . generally w, -^ . 



I shall prove this in the general case, by showing that if it is true of one value of n 

 (as we see it is), it is true of the next value, and so on. Let the {n — l)th substitu- 

 tion give 



d^q , dp , , X-Y , a-(n-l)(n-2) 



4 cos^ — - — 



and let 



dq , . ,, ^ X-Y 

 ^+ (w- l)f tan— g— = «: 



then, as before, 



d'-q , , ,- X— Y dq , n-\ 1 ds , 



dTdY + (/^ - 1 ) tan -2- '1^-^—2 rx=:Y * ^ = 5X ' 



and, therefore. 



ds , a — {n-\)n ^ , ds 4 „X-Y. 



dX + g ,X-Y = 0^ a^d ^ = -^ ^■,(;,_i);, C0S^-2-* 



4 cos' 



, X-Y . X-Y 



4 cos — - — sm 



dq _ d^s 4 ^ X-Y ds 2 2 



c?Y~ fiX<^Ya-(«-l)«^^^ 2 ~" rfX a-(w-l)« 



Consequently 



<^s 4 „X-Y ds 4 X-Y . X-Y 



cos-^ — t; -Tv ? tt:: cos — ;; — sin 



dXdYa-{n-l)n^^° 2 dXa-{n-l)?i 



ds 4{n--i) X-Y . X-Y _ 



"^ dXa-{n-l)n^^^ 2 ^^^ 2 ~ "*' 



or 



d^s 4 ^ X-Y , ds 4n X-Y . X-Y _ 



dXdYa-{ti-l)n^^^ 2 + d/X a -(«-!)« ^°® 2 ^^^ 2 +'^— 0; 



that is, 



d^s , rfs ^ X-Y . a-n{n-l) .X-Y ^ 

 rfX^ + ^ w tan —2- 4- * — ^ cos2 -^- = 0, 



and, therefore, the law of coefficients, as above stated, is correct. 



