INVERSE METHOD OF DEFINITE INTEGRALS. 



365 



To prove this property, conceive a curve Y 

 APC, of which the abscissa measured from 

 A along AB is taken equal to /, and the 



corresponding ordinate y is equal to f''', 

 and let us suppose h very nearly equal to 

 unity, and at any point P draw a tangent 

 PT; then since 



dt 



h 



y = t'-\ 



therefore, 11^ is the limiting value of the tangent of the angle PTB. 



Take AB = 1 and the ordinate BC = 1, then it is evident that 

 A and C are constantly points of the curve when the parameter h 

 varies so as to approach unity. 



Again, for the entire area APCB the expression is Ji^'~*, from t=0 



1 — A 

 to t=l, that is, - — Y, which evidently tends to vanish as the para- 



.« — fl 



meter k approaches unity ; and as no part of the area is negative, it 

 follows that the curve APC tends ultimately to coincide with the two 

 right lines AB, BC, and therefore when T is sensibly distant from 

 B the tangent of the angle PTB tends to vanish, but when indefinitely 

 near to B it tends to infinity ; and therefore Ug, which ultimately re- 

 presents these tangents, is zero from A to indefinitely near to B where 

 t is unity, when its value becomes infinite. 



In like manner the remaining functions C/i, U^, &c. may be dis- 

 cussed with similar results. 



It may be observed that for values of t>l (which however do not 

 enter the definite integral), the values of t/'^ are infinite. 



34. Expansion of given Junctions in terms of the functions Un . 

 The general formula for this purpose is 



0(0 = V^kW) + UJt<p{t). hA.it) 



1.2 



^,<t>{f).ih.\.tf + 



u. 



1.2.3 



.j;<^(0.(h.i. /)' + &c. 



3 b2 



