Misccflaiiea 107 



Wo olitiiin tlie fuHuwinj,' valuos 



By (1) Ist Ditroreiico.s (:5 valuea) 12-2549 



(1) 2nd „ (6 valucs) 12-2773 



(2) 4 values 12-2G19 



(4) Iß valuo» 12-2815 



The valuo giveii in tlie Tables i» 12-2811. 



In applving forniula' (4) if we usc 2» toi-nis in oacli equation \ve, require 4«- terms altogether 

 — thuM 4, IG, ;5G... are the nunibers of turni.s that will uc-cur. üf cnur.se if wo usc an odd 

 number of terms, aay 2« + l, the fcirnuila roads 



'2.. + 1 "-"+••• +-^ '+... and u„,r = r,„^A\'^+...+J-'ctc.\ 



«;-.r = P2.. + l| - +---+7_; + ---J- •^"" "7.:r = '-2,.+ lp- + ----*-— ^CtCj 



where p„a + i = (p + n) (p + n-l) ... (p+l)p(p-\) ...{p- n). The formula; can also be used 

 when m terms are used in one direction and // in the otlior. 



As an example of the füll working take the table of joiut life annuities given and find the 

 annuity on two lives aged 51-27355 and 28-63984. Here p = -25471, )-=-72797. 



log(ji) + l) -0985433 

 log(2-r) -1044973 

 log (2-/)) -2418676 

 log (»•+!) -2375362 

 log {p) T-4060460 

 log(l-?-) 1-4346168 

 log (1-ju) 1-8723253 

 log(r) T -862 1135 

 \og p^r^ 1-2575460 



'o = 2 ' '-'l = ^ 5' "^2 = iS' '' - 1 = ~ ff ' 



log 4 = 1-6989700 and log J = 1-2218487, 



and we get the following e.'cpressimis corresponding to {...} in (4), where [-2...] represents 



antilog -2.... 



"p" terms -[T-1233054] ';_i + [ -2929240] »„ + [1-8266447] »,-[2-9799811] Hj, 



";•" terms -[2-9843125] «_, +[1-836850.5] «„ + [ -2643532] »,- [T1173514] «j, 



( + ) 1-1520924 (-) 1-1075220 (-) 10512745 ( + ) -9845408 

 2-1076179 T-27723G5 2-8109572 r5-9642936 



(-) 1-1465311 ( + ) 1-1030747 ( + ) 1-0487757 (-) -9818277 

 2-9601619 -1297805 T-6635012 2-8168376 



(-) 1-1377117 ( + ) 1-0958316 ( + ) 1-0428827 (-) -9770602 

 1-3876586 -5572772 -0909979 T-2443343 



( + ) 1-1258878 (-) 1-0861960 (-) 1-0351735 ( + ) -9709788 

 2-2406568 T-4102754 2-9439961 2-0973325 



The ui)per line in cach ca.se gives the logarithni of the Joint life annuity, and the lower line 

 the sum of the logarithuis of the coeffieients shewn in the two expressions marked "/)" terms 



14—2 



