13] 
ON THE THEORY OF LINEAR TRANSFORMATIONS. 
91 
so that if U be transformed into 
U' = a'x' 3 + 3/3Vy + 3 ry'x'y'* + By, 
by means of 
x = X x' + p y', 
y = \x' + ptf, 
then if 
we have 
u' = a! 2 B' 2 - Ga'B'py - 3/3V + 4/3 ,3 S' + 4a'y, 
II = (\fl / — A,^) 6 . u. 
III. 
p — 3, m = 2, n = 4. 
Notation as in I. (C), 
u = A (& + 333 4- 3GD + 6B + 60)-5(<2D+B-3e-J+ 2ffi + 3^), 
where A, B are arbitrary constants, and 33, &c. ...p^ are functions of the coefficients, 
given as follows :— 
= a 3 p 3 — b 3 o 3 — c 3 n 3 4- d 3 m? — e 3 l 3 + f 3 k? + g 3 j 3 — hH 3 . 
33 = — a 2 p 2 bo 4- b 2 o 2 ap + c 2 n 2 dm — d 2 m 2 cn + e 2 Pfk — f-k-el — g 2 j 2 hi + hS~g] 
— a 2 p 2 cn + b 2 o 2 dm 4- c 2 n 2 ap — d 2 m 2 bo + e 2 l 2 gj — f 2 k 2 h% — g 2 j 2 el + h 2 i 2 fk 
— a 2 p 2 el 4- b 2 o 2 fk + c 2 n 2 gj — d 2 m 2 hi + e 2 l 2 ap — f 2 k 2 bo — g 2 j 2 cn + h 2 i 2 dm 
— a 2 p 2 hi + b 2 o 2 gj 4- c 2 n 2 fk — d 2 m 2 el + e 2 l 2 dm — f 2 k 2 cn — g 2 jfbo + h 2 i 2 ap. 
(if = + a 2 p 2 dm — b 2 u 2 cn — c 2 n 2 bo 4- d 2 m 2 ap — e 2 l 2 hi + f 2 k 2 gj + g 2 j 2 f k — li 2 i 2 el 
4- a 2 p 2 fk — b 2 o 2 el — c 2 n 2 hi 4- d 2 m 2 gj — e 2 l 2 bo 4- f 2 k 2 ap + g 2 j 2 dm — hH 2 cn 
4- a 2 p 2 gj — b 2 o 2 hi — c 2 n 2 el 4- d 2 m 2 fk — e 2 l 2 cn + f 2 k 2 dm + g 2 j 2 ap — h?i 2 bo. 
73 = apbocn — apbodm— apcndm + bocndm — elf kgj + elfkhi +elgjhi — higjfk 
+ apboel — apbOj k — cndmgj 4- dmcnhi — elfkap + elfkbo + gjhicn — higjdrn 
— apbogj 4- apbolii 4- cndmel — dmcnkf-\- elf ken — elfkdm — gjhiap + higjbo 
4- apcnel — bodmfk — cnapgj + drnbohi — elgjap + fkhibo + gjelen — hifkdm 
— apenfk + bodmel 4- cnaphi — dinboel + elgjbo — fkhiap — gjeldm + hijkcn 
— apdmel + boenkf +cmbogj — dmaphi + elhiap — fkgjbo — gjfkcn + hidmel. 
(¡B = apdmjk — bocnel — bocnih + dmapgj — elhibo 4- fkapgj + gjfkdm — hielcn. 
Jp —arphjo — b 2 goip — c 2 nflm+ d 2 emkn — e 2 dlkn 4- f 2 ckml + g 2 bjpi — h 2 aijo 
— i 2 phbg 4- j'ogah + ltfnfde — l 2 mecf 4- mHdcf — n 2 ckde — o 2 bjah + p 2 aibg 
+ urphkn — b 2 golm — c 2 nfip + d 2 emjo — e 2 dljo 4- f 2 ckip + g 2 bjpl — h 2 aink 
— i 2 phcf + fogde 4- k 2 nfah — l 2 mebg + m 2 ldbg — n 2 ckah — o 2 bjde + p 2 aicf 
4-a 2 phlm — b 2 gokn — c 2 nfjo + d 2 emip — e 2 ldpi 4- f 2 ckjo + g 2 bjnk — h 2 aiml 
— i 2 phde + j 2 gocf + k 2 nfbg — Pmeali 4- ni 2 dlah — n 2 clcbg — o 2 bjcf + p 2 aide 
+ a 2 pdno — b 2 cmpo — c 2 bpmn + d 2 aomn — e 2 hjkl 4- f 2 gilk + g 2 flij — h 2 ekji 
— Plfgli + j 2 kehg + k 2 jhef — Pigfe + m 2 pbcd — h 2 aodc — o 2 ndab + p 2 mbca 
+ (Aping — b 2 hkmo — c 2 ejpn + dfiom — e 2 cpjl + f 2 doik + g 2 anlj - h 2 bmki 
— Podfh 4- j 2 pceg 4- k 2 rnhbf — Pnage + vi 2 khbd — n 2 lgac — oHfbd + pHeca 
+ cPplfo — b 2 ekpo — c 2 hjmn + d 2 gimn - e 2 bpkl +f 2 aolk + g 2 dnij - h 2 cmji 
— i 2 ndgh + j 2 mchg + k 2 pbef - Poafe 4- m 2 jhcd - nHgcd - oHfab + p 2 keba. 
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