EEV. E. HAELET ON THE METHOD OE STMMHTEIC PEODIJCTS. 
339 
and, therefore, by Theorem III,, 
+ l‘i%cc\x^^X4C^ + i%x\x^^^x^ + ^%x\c(^^lx^ -\-12'%xlxlxlx^x^. 
Operating in the same way upon X®, we are led to the corresponding value of X®, and 
thence to SX^ Kepeating the operation upon we find ^X ^ ; and so on. But it may 
be remarked here that in dealing with the higher dimensioned functions, it will be found 
convenient to drop the subject x and to work only with exponents. Thus, following 
the first cycle and omitting, for convenience, the unit-sufiix, we have 
X=S'{(21001)+(20110)}=(21001)+(20110) 
+( 12100 )+( 02011 ) 
-}-( 01210 )+( 10201 ) 
+( 00121 )+( 11020 ) 
+( 10012 )+( 01102 ), 
and 
X=:^'[{(42002)+(40220)}+2{(33101)+(30311)} 
+2{(32210)+(32021}+(31202)+(30122)}] 
+ 8S42^+4S3^21+2S3^2"+2{S4B+S2n"}X. 
And the passage to Sx® is easily efiected. Carrying forward the calculation and collecting 
results, we have* * 
SX = 2S21^ 
2x-= 2S42^ + 12X41^ + 4^3^!^ + 42)32^1 +12X2T^ 
2x^= 2S63" + 2S62^1^ + 3X5421 + 6X532^ 
4- 8X532B+ 6X4^31 +12X4^2P +12X43^2 
+ 8X43*1^ + 6X432^ +48X42^ +24X3^21 
+12X3^2^ +2{X4r + X2^1^}Xx, 
XX‘‘= 2X84^ + 4X83^1^ +12X82^ + 4X7531 
+ 4X7432 +16X742^ +16^73^21 
+12X6^2^ + 4X6541 +20X6531^ 
+24X652^ +12X64^2 +20X64^P+24X643^ 
+28X64321+52X63^2^ +24X5M2 +24X5^4+ 
+48X5^3^ +16X5=*321 +72X5^2^ +44X54^21 
Thus if, in the expression l! x1ci^x\x\x\, we suppose {ex. gr.') that, of the five exponents, y is the greatest, 
then this function must he replaced by its equivalent H! x\x^^yx’lo(^. Or, suppose that the greatest exponent 
(y) is repeated, and that the function takes the form 'Sl x1o(^x\x\x ^ ; then this must be replaced by 
; and so on. Following this method, the comparison of similar functions is greatly facilitated. 
* Many of the details of calculation are given in the third section of my original memoir ; but the results 
there exhibited belong to the quintic wanting in its second, third, and fifth term. The results exhibited in 
the text belong to the perfect form. 
