CHAP. IX.] METHODS OF ABBREVIATION. 145 



premises) with respect to the following elements, viz., the pos- 

 session of red blood, of an external covering, and of a vertebral 

 column. 



We must first eliminate a. The result is 



ir { 1 - sn (1 - t) - st (1 - n) } + nt = 0. 



Then (IX. 9) developing with respect to s and /, and reducing 

 the first coefficient by Prop. 1, we have 



nst+ir(l-n)s(l-t) + (ir + n)([- s) t + ir(l-s)(l-t) = 0. (5) 

 Hence, if st = w, we find 



mo -f ir (1 - n) x (ir + n) x ir (1 - w) = ; 

 or, nw + ir (I - n) (I - w) = ; 



i 



- . ir ( 1 - n) - n 



= ra + r (1 - n) + Qi (1 - r) n + - i (1 - r) (1 - w 

 + (1 - 1) r + ? (1 - r (1 - n) + (1 - i) (1 - r) n 



Hence, soft-bodied animals enclosed in tubes consist of all 

 invertebrate animals having red blood and not naked., and an in- 

 definite remainder of invertebrate animals not having red blood and 

 not naked, and of vertebrate animals which are not naked. 



And in an exactly similar manner, the following reduced equa- 

 tions, the interpretation of which is left to the reader, have been 

 deduced from the development (5). 





