﻿On the Theory of Rectifiable Compound Logarithmic Waves. 375 

 7r__, 1_ _2_ _6_ n— n 2 *^ 



2~ + i+ i+ 1+ ■ i+» 



iifo. #r., when rc=4, -^ approximately equals 



. , 1 2 6 12 . 128 _ en . 



A + r+ tt tt r+i' '• e - =-8P or : 5803 ' 



352 

 and. when rc = 5, will be found to be ^r- or 1-5644. The un- 



corrected convergent corresponding to the former of these is, as 



we have seen, — , or 1*4222; and the next is — — ■ , or 1*7056, 



4rO «v<vO 



IT 



the true value of — being 1*5708. The errors given by the 



uncorrected factorial values are 'I486 and *1348 respectively (of 

 course with opposite signs), whereas the errors corresponding to 

 the corrected values are only *0094 and *0064; the approxima- 

 tion being thus more than fifteen and twenty-one times bettered 

 for the fourth and fifth convergents respectively by aid of the 

 correction. 



Athenaeum Club, 

 April 1869. 



LIV. On two remarkable Resultants arising out of the Theory of 

 Rectifiable Compound Logarithmic Waves. By J. J. Syl- 

 vester-]-. 



THE fruitful investigations in which I have been for some 

 time past engaged concerning reducible cyclodes and recti- 

 fiable compound logarithmic waves have led me inter alia to 

 notice a problem of elimination which from its elegance and 

 peculiarity is, I think, worthy of being offered in a detached 

 form to the Philosophical Magazine. 



* This comes to the same thing as saying that for the purposes of cal- 

 culation the continued fraction should be always considered as ending with 



1 13 



a numerator, 1, and not with a denominator, jr. Ex. gr. \-{- ■, , , , i, e.~n 



tt 1.4. 1 



is a good deal nearer to o-than 1 + , i. e. tt, is; and so 1-J- y , 1 



Q l 2 + 1, 



• • * 1 16 



or r, is much nearer to it than 1+ T i - , i. e. — -, is. 



b x +i+3 9 



By taking the mean between two consecutive corrected convergents, or, 



still better, the mean between two such consecutive means, and so on, a few 



terms will serve to give a very close approximation indeed to the limit |j. 



t Communicated by the Author. 



