280 PROFESSOR A. G. GREENHILL ON THE 



39. ABEL'S form in 'CEuvres,' 1, p. 142, 2, p. 1GO, for n = 2 becomes the same 

 as in 2, p. 148, by replacing his x + ^a by x, his a by 2 (q + q'), and b by 



1 / '\9 



-H?-?) 3 - 



But now in our notation for /A = 8, equation (14) 25 becomes 



Z, = [z + i (1 - m)] 2 - (1 -- 2m) 2 z (1) ; 



and putting 



1 + ft ft (I ft) 2 / 1 + 2a + a 2 \ 2 /.. 



m = ,! i = ' , 1 - 8m + 8m 2 = ( r - i (2) 



] _)_ 2a cf- 1 + 2 rt- 1 -f- 2a a 2 / 



and replacing z by * where M = 1 + 2<t 2 , the roots of Z = Z { Z. 2 = are 



(3). 



A quadric transformation is required in ABEL'S result to obtain our I (r), namely, 

 with the 8 above, and the sign of Z l changed to obtain the circular form, 



_ (1 +)"(! -)(1-O 



8 - re 2 ( >' 



_(l + rr)(-l + 2a + 2 )3 . 



Z ' ~- 3 " ( j> 



" 3 ~ 3 - x- 



and 



Z _ /, _ Z w 2 . _ - - - ( } 



" ^ " : '"^ (8 tc 2 1 2 



so that Y/Z., is rational. 



Afterwards the degree is halved, as ABEL'S integral is 21 (v). 



A similar quadric transformation and halving of degree will be required for all 

 even values of ABEL'S n, to reduce his results to our form, involving heavy algebraical 

 work. 



ABEL'S integral (' CEuvres,' 2, p. 148), changed into the circular form, can be utilised 

 for the construction of an algebraical orbit described under the central attraction 



