570 
ME. A. CAYLEY ON TSCHIENHAUSEN’S TEANSEOEJ^IATION. 
where it may be remarked that the greater part, but not all, of the component coejfi- 
cients are divisible by 5. I soon observed in the process of summing the ten dhisions 
that all the resulting coefficients should be divisible by 5 (the only exception is as to 
the terms in if which contain B®, C®, and E® respectively), and the cii’cumstance that 
they are so in each particular instance is as far as it goes a verification, which, however, 
only applies to those of the component coefficients which are not themselves dhisible by 
5. But it was known a priori (I will presently show how this is so) that the sum of the 
resulting coefficients should be equal to zero, and that they are so in fact is a verification 
as to (til the coefficients. The foregoing specimen term BCDE is one which remains 
unaltered Avhen B, C, D, E are changed into E, D, C, B ; and on making the further change 
«, 1), c, d, e, d, c, b, a, the coefficient of BCDE remains, as it should do, unaltered ; 
this is a verification of the coefficients of the terms ace^, Ifdf; ad^e, h(ff, which are 
respectively interchanged by the substitution in question, but not of the other terms 
d'f“, abef aedf Ife^, hedf, which are respectively unaltered by the substitution. I did not 
employ Avhat would have been another convenient verification of the seA^eral dmsions, 
viz. the comparison of their values on putting therein a—l)=c=.d=e=f=\, with the 
corresponding values as calculated independently from the determinant 
^_B-4C-6D-E, - B-5C-10D-10E, - C- 5D-10E, 
E, y- B-4C- 6D+ E, - B- 5C-10D 
D , 5D+ E, y- B~ 4C+ 4D+ E, 
C , 5C+ D , 10C+ 5D+ E, 
B , 5B+ C , 10B+ 5C+ D 
D-5E, -E 
- C-5D , - D 
~ B- 5C , - C 
y- B+ 6C+4D+ E, - B 
10B+10C-f5D+ E, ^+4B+6C+4D+E 
The calculation of the ten divisions of this determinant would however itself have been a 
work of some labour. 
The last-mentioned determinant is ; in fact, equating it to zero, we haA-e the 
transformed equation corresponding to the system of equations 
(1,1,1,1,1,11^., 1)^=0, 
y=:(^^_pl)B+(^2+5a.+4)C+(^=*-{-5a'^+10,r-f6)D+(a.^+5a'=’+10.r*+10a^+4)E. 
But the first of these equations is (a.-]- 1)^=0, and the second is 
y=(^+l){B+(.a;+4)C+(^^+4.r+6)D+(a;^+4^^+6^+4)E}, 
so that for each of the five equal roots 1, we have y—^^ or the transformed 
equation in y is y'‘=.^. 
