356 MEMOIRS OF THE AMERICAN ACADEMY 



(137.) **=0 J> 1 = 0, 



where 5 is less than unity and p is a limited relative. 



(138.) 



X 



1 = 1 1*=1 



In other respects the formulae for the two involutions are not so analogous as might 

 be supposed ; and this is owing to the dissimilarity between individuals and infinitesi- 

 mals. We have, it is true, if X. is an infinitesimal and X ' an individual, 



39.) X£y,9) = X t y,X t z like (jr^)X = yX,*X' ; 



(140.) X / ,y = X / ,Xy « X',y = X',yX'; 



(141.) (xj=i « m==«. 



We also have 



(14 



9 



-x> -< $ 



X „ v 



But we have not x t y = X t y, and consequently we have not s w -<^ sw, for this fails if 

 there is anything which is not a servant at all, while the corresponding formula 

 s w — <^ sw only fails if there is not anything which is a woman. Now, it is much more 

 often the case that there is something which is not x, than that there is not anything 

 which is x. We have with the backward involution, as with the forward, the formulae 



(143.) Ifz-O %?-<**; 



(144.) If x -<^ y z x — <^ z y . 



The former of these gives us 



(145.) ' i*w -< C)w , 



> 



or, whatever is lover to nothing but what is servant to nothing but women stands to 



■ 



nothing but a woman in the relation of lover of every servant of hers. The following 

 formulae can be proved without difficulty. 



(146.) 



l s w -< Zs w , 



every lover of somebody who is servant to nothing but a woman stands to nothin 



but women in the relation of lover of nothing but a servant of them 



(147.) 



l sw —^ l 





or, whatever stands to a woman in the relation of lover of nothing but a servant of 

 hers is a lover of nothing but servants of women. 



