496 Sir W. R. Hamilton on some 



is, from the polynome P, and may write more briefly (as in qua- 

 ternions), 



1 = S + V, K=S-V; (21) 



\trnpricp 



S=i(l + K), V=i(l-K); . . . (22) 

 or more fully, 



SP=|(P + P'), VP=i(P-F), if P' = KP. . (23) 

 Again, since (with the recent meanings of ^ and it), 

 Sp=p, Yp=.0, Kp=p, Sct=0, V'Gr='sr, K'cj=— ■z<7,'l 

 S{p — 'ar)=p, V(/; — ot)= — OT, K(/)— ■57)=^ + 'cr, J 

 we may write 

 SSP=SP, VSP = 0=SVP, VVP=VP, ~l 



SKP=SP = KSP, VKP=-VP = KVP, KKP=P;J * ^ ^ 

 or more concisely, 



S2 = S, VS = SV = 0, V2 = V, 1 



SK=KS = S, VK=KV=-V, K2=l. J " \ ,4 

 The operations S, V, K are evidently distributive, 



SS=2S, VS = 2V, K2 = 2K; (27). 



and hence it is permitted to multiply together any two of the 

 equations (21) (22), or to square any one of them, as if S, V, K 

 were ordinary algebraical symbols, and the results must be found 

 to be consistent with those equations themselves, and with the 

 relations (26). Thus, squaring and multiplying the equations 

 (21), we obtain 



12=(S+V)2=S2 + V2 + 2SV = S + V=1, - 

 K2=(S-V)2=S2+V2-2SV=S + V=1, l. . (28) 

 1K=(S+V)(S-V) = S2-V2=S-V = K; J 

 and the equations (22) give similarly, 



s«=i(i+K)2=^(i+K2+2K)=i(i+K)=s; -^ „';,;^ 



V2=,}(l-K)2 = ^(l + K2-2K) = i(l-K) = V; I {29^. 



SV=VS=i(l+K)(l-K) = i(l-K2)=|(l-l)=oJ "' 

 Again, if we multiply (22) by K, we get 



KS = iK(l+K) = i(K + K^) = i(K + l) = S, . 1 

 KV = iK(l-K) = i(K-K^) = i(K-l) = -V;/ * ^ ' 

 all which results are seen to be symbolically true, and other veri- 

 fications of this soi't may easily be derived, among which the 

 following may be not unworthy of notice : 



/l + KN"' 14-K 

 (S±V)2"' = 1, (S + V)^'»+' = S±V, {~^) =^. • (31) 



where m is any positive whole number. 



