expressed by a New Formula, 9 



pared, of A log P. By the aid of this property, Tables of mor- 

 tality may be constructed for all ages without direct use of the 

 new formula, except for determining one value of A log Y t for 

 a particular age (t) at a known distance from the zero of life. 

 When one value of log A log P* has been obtained, all other 

 values of log A log Pf may be obtained by successive additions 



or subtractions of the quantities extracted from a Table of 



reciprocals. The property mentioned will be found illustrated 

 in Table IV. hereunto annexed, wherein the unit interval of age 

 in the period of manhood has been assumed to be 6 years. 



The distinguishing property, just mentioned, of the new 

 formula is dependent on, and deducible from, the law which 



connects together the differential coefficients * ° and the 



logarithms of such differential coefficients. It has already been 

 shown that such differential coefficients are represented by 



<■=•(' +0 



Taking the ratio of two consecutive differential coefficients at unit 

 intervals, 



a t+l = d.logV t +i = _/ a + t + l \-'l 

 a t ' d , log Pj , V 0-fc t / 



and taking com logs of both sides of above equation, 

 com log ei t+l — com log at= — - icom log (a + 1-\- 1) — comlog (a + 1) \ 

 = - {hyp log (« + / + l)- hyp log {a + t)\, 

 the above equation also gives in hyp logs, 



nyp log ** = - J hyp log ^±1 = _ | nyp i og ^ + J_) 



1 f I 1 1 „ \ 



~ k\a + t 2(a + t)* + 3(«+*) 3 'J 



= — T x at the limit. 



k a + t 



That is, when the intervals of age are indefinitely small, the 

 differences of the hyperbolic logarithms of the differential coeffi- 

 cients of log Pj are = — r x —7—. t and consequently the differ- 



K CL-\- t 





