84 MR. C. J. HARGREAVE ON THE CALCULATION OP ATTRACTIONS, 



n — -. — F 



4 M 9 



COS* 



Y - X — 3 Y - X 



+ 'lL+l)e"tan»-2^+.... 

 . . . + "Ji + 1) c(» - 1) tan ^ + "^> c(») 



Whence we obtain c' = 0, c'" = 0, c^ = 0, &c. 

 Also 



/"^i^+D _ (/ »-l)(yi-2) \ _ 2»-3 / /i(w+l) (n-3)(/^~4) \ i^ _ ( n~3)(n-2) „ 

 \4 4 /^— [;i_2]'V4~4 /^~ 4 ^' 



/^^ (^ + 1) _ (>^ - 5) (y^ - 6) \ ^^ ^ (^-5)(«-4) ^iv ^ ^^^ ^^^ 



Consequently 



P = c 



Now 



iw — 1 



+ 

 + 



tan' 



Y-X 



+ 



in-l 



2 ^ 2[w-2](2n-l) 

 2«- 1 



tan 



»-2 



Y-X 



2.4. [w - 4] (2 « - 1) (2w - 3) 



tan 



n — 4 



2 

 Y-X 



}«-^ tan"-6 



Y-X 



2.4.6.[«-6].(2w-l)(2«-3).(2«-5) 



+ . 



Y-X 



= — 2N/-ll^gr^=tan-^ (-^x/-l),.. 



tan 



Y-X 



(-.y^V'~l)«-4 



-l)(2n-3)^"*/^ 



2 2 V ' — ° 1 — /A 



whence, finally, 



P ^^n-x^ a-i^^^^r ■ (-/.^•:ii)«-2 



V [w] 2.[«-2].(2w-l) ~ 2.4.[»-4](2w-l)(2n-3) 



a remarkable result, showing that in this instance P^ is independent of (p. 



P being free from <p, i^ a perfectly symmetrical function of ^ and ^ ; and ^ is a 

 constant with respect to ^ ; therefore 



*^„ — *NV [„] "'"2 [w-2] (2n-l) ^2.4.[»-4] (2w-l) (2w-3)^"V ^ 



V [w] ^ 2.[w-2](2n-l)^ ""/* 



To determine K^ for any particular value of w, we refer to the expression from 

 which the two series were deduced ; namely, 



|y.2 _|_ y.'2 _ 2 rr' [y.^ '\-\/{\- [ju^) ^(1 - f/.'^) cos {(p - <p')) | ~ ^• 



When jO. and jO*' are each 1, then P„ = 1, which gives an equation to find K^. 



