136 MR, JAMES COCKLE; RESEARCHES 
’ Sah AOS QB Oi pie,  eaeguy x (13), 
where 
108AB_ A 
C= ae ag ee ee eee (14). 
§ 16. 
Determining A by the equation A=B)’, and assuming 
AM=B)M=5H, we obtain from (14) 
c2— 108(Ax') (BA) par Andt 
52,0 5°(BAS)A” 
or 
Cr®=(Cr y= 108 H*— li*. 
Consequently, when (14) is satisfied, (13) may, by means 
of the assumption f=A%, be transformed into 
®45Hf+,/108—H-Hi+5H=0..... (15), 
or F 
{@4-6H(P+1)}"=(08-H) AH? 2S (16), 
or, making ?=s, 
di? POEs 4 1)g*= (108) H's *.' 5 (17). 
§ 17. 
Tf, therefore, we determine T and F from the equations 
ig! by Qe § ard is 
which in fact lead to 
PS pts, 
the solution of (17) is made to cat upon that of 
Spe om =O" aes swank (18), 
a form into which the assumptions 
FPO wt=E, pé=u 
suffice to convert (1). 
§ 18. 
The following remarks arise out of this discussion : — 
le. The functions a, 6, c and d seem to be entitled to 
rank as canonical expressions in the theory of equations 
of the fifth degree. 
