434 Rev. H. E. C. Logan on the Calculus of Factorials. [June 18, 



4. On comparing the action of such compounds as C 9 H 7 N (chinoline) 

 with C 9 H ]3 N (parvoline &c.), or C 8 H n N (collidine) with C 8 H 15 N (conia, 

 from hemlock), or C 10 H 10 N 2 (dipyridine) with C 10 H u N 2 (nicotine, from 

 tobacco), it is to be observed that the physiological activity of the sub- 

 stance is, apart from chemical structure, greatest in those bases con- 

 taining the larger amount of hydrogen. 



5. Those artificial bases which approximate the percentage composition 

 of natural bases are much weaker physiologically, so far as can be esti- 

 mated by amount of dose, than the natural bases ; but the kind of action 

 is the same in both cases. 



6. When the bases of the pyridine series are doubled by condensation, 

 producing dipyridine, parapicoline, &c., they not only become more 

 active physiologically, but the action differs in kind from that of the 

 simple bases, and resembles the action of natural bases or alkaloids 

 having a similar chemical constitution. 



7. All the substances examined in this research are remarkable for 

 not possessing any specific paralytic action on the heart likely to cause 

 syncope ; but they destroy life either by exhaustive convulsions, or by 

 gradual paralysis of the centres of respiration, thus causing asphyxia. 



8. There is no appreciable immediate action on the sympathetic system 

 of nerves. There is probably a secondary action, because after large 

 doses the vasomotor centre, in common with other centres, becomes 

 involved. 



9. There is no difference, so far as could be discovered, between the 

 physiological action of bases obtained from cinchonine and those derived 

 from tar. 



XVI. "On the Calculus of Factorials." By the Rev. H. F. C. 

 LOGAN, LL.D. Communicated by Professor CAYLEY, F.R.S. 

 Received November 10, 1873. 



(Abstract.) 



Our present knowledge of what is called pure analysis has for its con- 

 crete basis the general theory of powers. 



This science the author might, after Wronski, sanctioned by Lagrange, 

 have called algorithmic, but he prefers giving it the designation Calculus 

 of Poivers. 



The simple functions whose properties and relations it is the object 

 of this latter calculus to determine are, first, the three direct functions 

 or algorithms, z n , a 2 , sin s ; secondly, their three inverse functions or 



algorithms, z (or &z), log a z, sin" 1 ;?. 



The author proposes to establish a new branch of analysis or algo- 

 rithmie, which is based upon the general theory of factorials, and in 

 which z }l /? A * replaces z n . 



