VOL. LXIX.] PHILOSOPHICAL TRANSACTIONS. 483 



appeared, even after the water had been drawn ofF, as large as a woman in the last 

 month of pregnancy. It would have added greatly to his satisfaction to have 

 been able to clear up this point in every particular, by opening her after death ; 

 but he had the extreme mortification of being denied this necessary circumstance, 

 notwithstanding his most earnest solicitations. He was therefore obliged to 

 content himself with giving this bare recital of facts, which will serve to record 

 to futurity a case, which he believes has not its equal in regard to the number of 

 operations. It is remarkable that this young lady had a good appetite for the 

 most part, and was very chearful ; and, except a day before and after each ope- 

 ration, used to visit her friends at several miles distance as she would have done 

 in health, and till within the last 2 or 3 months could walk a mile or 2 with 

 tolerable ease. 



As to the quantity of water drawn off, Mr. L. found it to amount to about 24 

 pints at each operation ; for though the first time produced only 12 pints, and 

 in several of the latter operations the quantity fell short of 24 pints, yet he 

 might venture to state it at least at 24 pints or 3 gallons on an average, as in 

 many of the operations he took off from 28 to 30 pints. The number of times 

 he tapped her was in all 155, which brings out in the whole 3720 pints, being 

 465 gallons, not far short of 7-s- hogsheads. 



VIL Problems concerning Interpolations. By Edw. IVaring, M. D., F. R. S., 

 and of the Institute of Bononia, Lucasian Prof, of Mathematics, p. 59. 



Mr. Briggs was the first person it seems that invented a method of differences 

 for interpolating logarithms at small intervals from each other: his principles were 

 followed by Reginald and Mouton in France. Sir Isaac Newton, from the same 

 principles, discovered a general and elegant solution of the above-mentioned 

 problem: perhaps a still more elegant one on some accounts has been since dis- 

 covered by Messrs. Nicole and Stirling. In the following theorems the same 

 problem is resolved and rendered somewhat more general, without having any 

 recourse to finding the successive differences. 



Theorem 1. Assume an equation a -\- bx -^ cx^ -\- dx^ .... x"- ' = y, in which 

 the coeificients a, b, c, d, e, &c. are invariable; let a, j3, y, ^, $, &c. denote 71 

 values of the unknown quantity x, whose correspondent values of j/ let be repre- 

 sented by s', s'^, s^ s', s, &c. Then will the equation a + bx -{- cx"^ -f- dx^ + 



X 



1—1 





, ^ — M..^—^,.a — »■■>-. .»v.. y , J-«-J — /3-X— y.x — t.&C . 710 



1"^_«.y_/3.7— .J'.V-i.&C. "^ ' ^—c^.i'- fi.i'-y.i'—i.Scc. '^ ^ + OCC. 



Demonstration. Write a for z in the equation 7/ == 



3a 2 



