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178 Professor- Baker, On the transformation of the equations of 



so that, whatever a, h, c, d, a', b', c', d' may be, we have 

 {e^a + e^h + e^c + e^d)^ = 0, 

 {Cja + e,^ + ggC + e/^d) (eia'+ e^'+ e^c' -\- e^d') 



=^ — (eia'+ ^2^'+ 630'+ e^d') {e-ia + gg^ + 63C + e^d), 



while 



{eja + e2b + eQC + e^d){eia'+e2b'+e^c'+e^d') {e-^a"+ e<^"+ e^c"+ e^d") 



in which A^, A2, A3, A4 denote the determinants obtained from 

 the array 



a, 

 a\ 





d 

 d' 



i 



II 'LI I ^11 Jll 



a , , c , a 



by omitting in turn the first, second, third, fourth columns, and 

 prefixing respectively the signs +, — , +, — . 

 We shall put 



^1 = 636364, E2, = 63^1^4' -^3 = e^e^e^, ^4 = - ^162^3 

 and shall also introduce the four units given by 



(e/, 62, 63', 64') = p' (ei, 62, 63, 64), 

 so that 61'= :?>ii'ei + ^12^2 + Pi3^3 + Pu'^i^ ^^c. 



These equations are equivalent with 



^1 = :Pnei'+ ^512^2'+ Pi3^3'+ Puei, etc., 

 and we may write the relations e'= p'e, e = pe'. The four com- 

 binations 



77' „ I I I T? I „ l„ I „ I T? I ^1^1^' T? ' „ '„ '^ ' 



hj^ = 62 63 64 , ^2 = 63 61 64 , 2^3 = 61 62 64 5 -£^4 ^ - «1 ^2 «3 



are then hnear functions of E^, E^, E^, E^; we find in fact 



(^/, E2', E^, El) = m-^p {E„ E2, E^, E,). 



The laws of transformation from (e^, ...,64) to (e^', ..., e^) are 

 the same as those from {dx,...,dt) to {dx', ..., dt'). Thus the 

 equations 



m'= pmp, n'= pnp, 



lead to m'e'2 == me^, w'e'^ = ne^, 



where as usual me^ is the notation for the quadric form 



^ 1^1^26364) (61^26364)' ^ 2jiu.j.gefeg. 

 For instance 



me^ = m (e) (e) =: m {pe) {pe') = mp) {e) . j) {e') = p . mpt {e) {e) 

 - m' (e') (e') = m'e'^. 



^ 



