418 PHILOSOPHICAL TRANSACTIONS. [ANNO 17Q8. 



H,L * l ( 8 i "l6J( = i6 H ' L,a ( 3.5.7.3 3 . 5 ._7 . 9_. ,5_ 3.5.7.9.11.7 



1 A_l_£_ X —iZlV" 4. Z£.( 4.6.8.4. 8~»~4 . 6 .8 .10 .6 . 4 ' 4 .~6 . 8 . 10 . 12 . 8 . 6* 

 *"l2'8 32 192 '192' 



&c. which is the value of another series of the preceding paper. 



5. If the last literal equation be divided by y, and ^pjJLipL = l^l be then 



taken from both sides, we shall have 



,^_5 _ 7__ _5_^ , ( 3_ 



M2^ 7 "" 12/ l6y*' ' U ^8/ 



, _5 , _3_ 1 67 



"' 12m 7 "V RuS 



4.6.8.4.2 512 



123/ 7 ' 8/ 32/ 1923/ 512 



3 . 5 . 7 . 9 ■ 5/ , 3.5.7.9.11 .7/ , 3 . 5 . 7 . 9 . 11 . 13 . 9 / - 1 • , 



4 . 6 . 8 . 10 . 6 . 4 "^ 4 . 6 . 8 . 10 . 12 . 8 . 6 "*" 4 . 6 . 8 . 10 . 12 . 14 . 10 . 8» wniCtl 



equation, in fluxions, gives 



q V 12y 9 ' 12/ 48/ l6/' •* v 8/ ' 163/*' 



"■"■^ 12/ 8/ 32/ ' 192/ 512' ~ 



3.5.7.9 -5.3jy , 3.5.7.9.11 .7.5}? 3.5 .7-9.11 . 13. 9. 7jy 5 « 

 4 . 6 . 8 . 10 . 6. 4 "*" 4 . 6 . 8 . 10 . 12 . 8. 6 " "*" 4 . 6 I 8 . 10 . 12 . ,14 . 10 . 8 ' 



Now the fluent of the fluxionary series on the 2d side of the equation being obviously 

 thP spHp* 3 . 5 . 7 . 9 . 5 . 3y y , 3.5.7.9.11.7.5/ 3.5.7.9.11-13.9.7/ 



inc & cnc & 4#6#8>10#6 . 4i2 -r 4 .6. 8 . 10 .i2.8.6.4" r 4.6.8.10.12.14.10.8.6' 



&c. we are next to take the fluent of the expression on the first side, and to 

 correct it, that it may be = this series; which may be done as follows: 

 For the fluent sought, assume 



z^ + j.+ ^+j,) +«($+%+>>) + f< + j; + j.+^ + '»-^> 



and take the fluxion of this expression ; then these fluxionary terms being put = 

 to those on the first side of the preceding equation, there will arise several equa- 

 tions for determining the values of the letters, a, b, c, &c. and then the assumed 

 fluent becomes, 



( -35 95 75 105 . . 9 , _5 , 105 > 



° *8 . 12/ 12 . l6y 6 4 . 8 . 8/ ~~ 8 . 8 . 83/5/ "*~ U ^32/ ■" 323/3/ "*" 8 . 8 . 8' 



-I „ + -7T- * — - — - — — - *, which may be corrected in the 



~ 8 . 12/ l6y« 4.8. 123/3/ J 



manner shown in the 2 preceding articles, when it becomes 



( - 35 95 75 105 >, , (9_i__A ■ 105 v 



tt ^8 . 12/ 12 . 16/ l6 . 16/ 8.8. 83/3/ "" U ^32/ -+ ~ 32yy "*~ 8 . 8~ . 8' 



35 5 # 37 1023 _ , 



+ 8~T27 8 + 16/ 4.8. 12^/v 4096 "~" ttie SenCS 



3 . 5 . 7 . 9 . 5 . 33/3/ , 3.5. 7.9. 11-7. 5y^_ , 3 . 5 . 7 . 9 - 11 • 13 ■ 9 • 7/ & 



4 . 6 . 8 . 10 . 6 . 4 . 2 ' 4 . 6 . 8 . 10 . 12 . 8 . 6 . 4 "* 4 . 6 . 8 . 10 . 12 . 14 . 10 . 8 . 6' 



which, when y = 1 , becomes 



9 ,5 .105 



H.L.2.(- +- +— 8 8 )| ^ 



.35 .5 37 1023 f — i 



"• 8.12M6 4.8.12 4096 J 



'2288 ' 512 



