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



Such is not the case. At each integration a function of X must be added, and these 

 functions determined by reference to the original differential equation*. 

 7. Returning to the value of P^, we have 



Po = X, (f - I- ^^T log \±t)+^{p + ^ ^—l log \±0. 



Now in the calculation of attractions, where P© is the coefficient of r^ in the ex- 

 pansion of 



{r2 + r'2 - r r' ((j,[l' J^ ^ {I ^ fj,^) ^ (l -^ (/,'^) cos (<p - 9')) } ~ ^' 

 we know that it is 1 ; consequently 



X, (, _ iy^log^) + ^(^ + ^v/^log^) = 1, 

 and expanding by Taylor's theorem, we get 



x>^-^.? i-v/^iogi^ - ^ a-iog^^^y 



+^,+^',iy^iog^-^(i-iogi±^y 



By equating the coefficients of the same powers of-^- log ^ __ , we have Xi ^ -\- "4^ ^ 



= 1, and -4/' (p — x'l ?> = O5 or -v)/ 9 — Xi <P = constant. 



Therefore -v// (p and Xi "P are absolute constants, and their sum is 1 ; whence it fol- 

 lows that % (p = 0. Let -4/ </> = g ? then 



> =1. 



'n = ^ ( • • • .y cos- 



_2Y-X 



^cos 



-2Y-X 



/ 2 



cos 



-2Y-X 



dYdY ...(n times)). 



2 ./ ^"" 2 ../ 2 ^"" 2 



Effecting these integrations, we find that P„ consists of a series of powers of 



* The common differential equation (1 — /x^) ^li> _ 2 u^ + ra (« + 1) P„ = will illustrate this point. 



a fi^ a fi 



Let P„ = ^~^, and after substitution, differentiate n times ; then (1 — m^) -t-? — 2 (ra + 1) /x -— ? = 0, whence 

 dfi-n dfi^ an 



z s= /* -i^ — -j- m. It is clear that no more arbitrary constants than k and m can be introduced ; and 



^(l_^2)n+l 



yet if the integrals were left indefinite, we might obtain an integral of an expression which should differ from 

 the integral of the same expression obtained by a slightly different process, by a constant. By another inte- 

 gration this would cease to be a constant, and we should obtain thus different values for Pn. The fact is, that 

 constants must be added at each integration, and recourse had to the original equation, to determine them in 

 terms of m, k, and fx. 



