FLUXIONS. ooo. 439 
x aha $25) constant, but indeterminate co-eficients, and let us ass Inverse 
—2ae * sume. Method. 
Inverse 
Method. Wehr i 116, Ex. 6.) that 
—— 
4 a dz Kgorrrio y Lae —— 
= 24s dx - dx , dx. , 
Eh + —l —¥{ 1): =e (2*+,°)" “@+a—) et @4ey-. 
cake Clarita ) fhe’ Let the uxions be. ‘iathag a which, in. respect.to 
. quantities affected by the sign f; is done by merely res 
+ ah = ahaa 4 the mon 
sGtiy jecting the sign, and we have » 
ale +1) +6 dz. Kdz 2K (wee ste 5 
Case IIL. ‘The fraction Art dz may be ak @+Fy~ +e” (eee 
+ 
— "“Azdz Bdz The fl en Bak tie yas" 
ved, inte faa mae 37 e fluent of By reducing these fractions to keting sl 
of these is 4 Al. (2* + 6*) +c, (art. 114 Rule III.) ba and rejecting such : ape as are found in all 
Hs B oz), the terms, we get 1=K (2 + 41) —2K (11) 2? 
and the fluent of the second is [are (tan. = =)+e +L (2 + 62), or : 
(114, Rule V,) therefore, uniting the corrections of the { (2n—3)K—L ple K+ L) 0; 
two 
Hence, by the theory of indeterminate co-efficients, 
var Didee FALQ 489 4 > are (tun. = = =) +e. (2n—3) K—L=0,1—47(K + Lae, 
Wa have found (alt. 116.°Ex., 5.) that From these equations we find K = = (n—1) 6” 
adz righ aie x (®—1)dz Qn—3 | ; . 
Aoi si Pa eT yeh L= sia —1)eF Therefore, substituting for K and 
The fluent of the first term of the second member is L, we find 
Lie (art. 114. pimp Make z= z— } by dz Bon 
d«=dz, then Ap separ termacia tramefermntad to eccemenrice: tak saat + 8°)" 
, zdz 
tT +3 Qn—3 ff dz 
Now epee ‘ELD; tea—pe/ G4 
(art. 114. Rule III.) and the fluent of the other is Te ey Joitee the woe, sf Se Ps by means of 
37S are (t= 75) therefore, restoring x in these it the fluent aay is made to depend upon ano= 
expressions, we ae 
adx * f capppeein which pia diminish y an unit, 
3.1? fug@enyery (tyes yt This last, last, again, is reducible to another, in which the _ 
¢ ; ‘ef @e41 exponent ee and so on, until we come to the 
te sare (ton = =F) fuent of ~ 
bringing ing the logarithmic i i into qne.term, MM AHoS 
a epostg the arbiter correction to be 41. (c’), The tain ERE Ee will serve to 
abl 1) rau ne ew ey exemplify this Peel 6 5 decomposed it becomes 
L {<< team um. = are (tan. = ae ) (—2e+41)dz2, Pci +1)dz dx 
v (#4 4-2-1) 73 (#1 epi * #41 
As a second example, veers DE The first terms of each of the two first fractions give 
1) (#41) 
= bes 116, Ex. 4.) may be decomposed ) REET 6) Vg hin Pde, = 
into 
@ +1) 2@o4 @piy= = eopT 
me pens, the latter of these’ again may With respect to the other terms, ba,sny, feral, 
adx > : dx . 
teem tei tei Teme | fears ate t Yee | 
ae Sie dan eh seh teeta of ovr 
‘ ry a clo de ha 3 . second eran! we have in like 
ay aside pabivnenaemaiand 
z+ dz Azdz z a : a 
matey The fluent of the first of eho ie, E309 oi arin seein’ tasty Sta 
ay eaters, email 
2 (mn — 1) (# BER Mrs 5 Ue Heloll) Bat it The ditierent parts apited.give ns 
rth WA Piles ath as ip hay re (tam 2) +0 
+ + 
120, Tofind the fuent of Trzgop let Kand Lhe FFI) ae hue 
3 which is known by Ritle V. art. 114. 
