76 Dr Baker, On the Invariant Factors of a Determinant. 

 see in fact that p*<> a -» +1 divides by ©«, that P"*- 1 * 2 divides by 



a« ^ ' ok J 



6 2a , and so on, and finally that P KCL divides by ©**. 

 Now consider 



Q = -Pax + -?W + • • • + Py<T ; 



if we take a power for which there is one term of its expansion 



(r + s + ...+t)\ pr ps , 



r ! s ! . . . t ! aK &*• w 



in which every exponent r, s ... t is less than a significant exponent, 

 this power is not a significant one ; thus the lowest possible 

 significant exponent is 



«(o-l)+>(/8-l)+... + «r(7-l) + l J 



or I — (k + fi + ... + o-)+ 1, 



and conversely this power divides by the lowest of the powers 

 © a , ® 3 , ... ®y, that is by ©*. The highest significant exponent is 

 similarly 



Kd + ft ft + . . . + 0-7, = I. 



We have in fact, as is immediately clear on consideration, the 

 following statement of the powers of (D by which the various 

 powers divide 



Qi-(«+H-...+(r)+i by©*, ..., Q*-<«+f*)+i by &nr+~+fi > 



Ql-K+l by @<ry+...+rtS+a 



Q?-(it+M+...+(r)+2 by 2 ?, ..., Qi- («+/*) +2 by ©o-y+-+2/3 j 



Qi-K+2 by (H)<ry+...+rt3+2a 



n?-cc+^+...+<r)+<r by ©°-y ? ..., QI-(k+h)+p by ©o-y+-+M^ j 



Q« by ®ffy+...+rt8+««. 

 Lastly consider 



where Q' involves ©' = p — 6', and Q" involves @" = p — 0", and so on. 

 The lowest significant power must be one such that in every term 



(r r^Mr"L.7 )! <y(Qy(0 " r - 



of its expansion, that does not vanish, the power r is a significant 

 power for Q. As the lowest vanishing powers for Q\ Q", . . . are 

 V + 1, l" + 1, ... we see that for the significant powers of R 



r + r -)-r + ... 



