380 Rev. B. Bi-onwin on the Rectification of the Ellipse. 



■d^y cP-y difl dy d*v d'y v'^ dy 



d X- d V- d X- dv d x' dv- x'^—\ dv 



^ . -: and the equation x^ — \ , -—— -\ v = 



^1 — ij- — 1 * d x^ 4 



changed into v"-- 1 — v- . -— ^ + 2v -~- -\ — - (l — if)y=o. 



O dv- dv 4 ^ ) J ^' 



_X 13 2 7. 



Assume y = av - + av'^ + av^ + &c., and substitute for 

 '— ^, -— , and y in the preceding, and we shall find a = 



dv" dv "^ ' ° 



1 2 \ 3 3^ 4 3*. 5^ 



-^a,a= -577^ a,a = 2-. 4.. 6' «? a = "^TTi'^'e^rsr «j where 

 a is found = 1. This gives the elliptic quadrant = 



+ &c. ; which is the first series in the place above referred 

 to in Leybourn, the inventor of which is Mr. Ivory. 



Next let us consider j/ a function of « =—^-r = c- = square 

 of semi-conjugate. Then the equation will become m(1 — u)^ 

 4^ - 2m (1 - M) ^ +±y = o. Making «=««'" + 



du- ^ ^ du 4 ■^ ^ ^ ' 



au + &c., we find m = 1, or = 0. But in the second case 

 a also = 0, and the two series become the same ; viz. y = 



au + au- + aii^ + &c. ; whence we obtain a = (2 — T") 2' 

 1 2 1 



2/ l\a 2a 3 / l\a 60 4 



« = V8 - -^J-Q - -g-, a = (^IS - jj-^^ - ^^, a = 



3 2 



^32 — -j)-^ ^> &c. where the law of continuance is evi- 

 dent, and a remains arbitrary. This series is to be multiplied by 

 ■|- = -- a/ 1 — u; hence it becomes nothing both when 



« = 0, or M = 1 , that is when e = 1 and e = 0. It must 

 therefore only be regarded as a particular solution ; it will, 

 however, be useful in seeking the complete one. 



Make y = P + g log. Cu. Then substituting this value of 

 y and its differentials in our equation, and making the part 

 multiplied by log. Cm separately = 0? we shall have the same 

 equation in q as we had iny; consequently the value of ?/ just 

 found may be taken for that of j ; we shall have moreover 



