4l6 PHILOSOPHICAL TRANSACTIONS. [ANNO 1798. 



in the same equation in which it occurs, and consequently that they are no longer 

 to be considered as part of that equation. This I have found to be better than can- 

 celling, as it answers the same end without obliteration. 



3. As it does not seem necessary to set down the operations of computing the 

 sums of all the series which arose in the preceding paper, I shall make choice of 

 the summation of those which, being the most difficult, are the most proper ex- 

 amples to illustrate this method. 



It is well known that the expression 



2j,r s . _ • 4 , g££ , 2- %"' . 2-3-% , 2.3.5.7 jy» , 2 . 3 . 5 .7.9jj/ 5 



^(1 _ yy ) li> — ^2 "T 2 T 2 .4 T 2 .4.6 T 2.4.6.8 "»" 2.4.6.8. 10* 



&c. from which equation we have 



jizryyj - *yy -yy — 4 - 4 .6 + 4.6. 8 + 4.6. 8 .io' * c - ^ ^ 



the fluents of the terms on 2/ 



the 1 st side are | 4. _L _i L 



[ ~ V 2yy 



on the 2d side, the fluents are |^| + 4 'g'g^ + ^[e'l.io ^ &c * And ' to find 

 whether these 2 expressions are = each other, or have a constant difference, we 

 may compute their numerical values, y being put = any small simple fraction, such 

 as -rV> i-5-5-> or \ o'oo ? either of which values of y is a very convenient one for the 

 purpose. But an easier method to discover the constant quantities which lie con- 

 cealed in some of the terms on the first side, is to convert that side into series, by 

 the binomial theorem; which will then be as follows: 



— V( l — yy ) , _4 1 , -* 1 , 1 1 ,.2 4. 5 -/ & r 



2^ — Try Ttj/ TttT -r^y n~ -a-mry 9 o"" 



~3\/(i — yy) __ 3 _2 1 3 j_ 3 7 ,2 1 9_ w 4 fc. c 



4— — — t# T t T- ttt3/ T Try J «C. 



+ 4H- L. i+^.yy) = -4H.L.2+-ry/ + -rf- s y, &C. 



The sum is = * * + T V — i H - L - 2 > + iVy* + tA&^S &c which evi- 

 dently differs from the series on the 2d side by the constant quantity -fo — ^h. l. 2. 

 We therefore have, by subtracting this constant quantity from the first side, 

 jv(j -yy) 3V(i-yy) , 2 } 



2<y 4yy Tt * * 1 + <•(! - yy) ( 3.5yy 3.5-7^ , 3.5.7-9/ . 



1 _L i J_ _ _L I "" 4.6.2 "*" 4.6.8.4 "r" 4.6.8.10.4' 0CC * 



"t" 2j/+ ' 2yy 16 J 



which, when y becomes = 1 , becomes 



# * 



I+4- + 



L +4-H . L. 2 1 3.5 3.5.7 , 3.5.7.9 & 



4. — ^ I -{- -jP T J = = 4 . 6 . 2 + 4.6.8.4~ i ~4.6.8.10.6' 



4. If the equation of fluents in the preceding article be divided by y y and if 



— J£ = 4 V be then taken from both sides of it, and u be written for h. l. 

 4.6.2 16 •? 



