. Differential and integral calculus. e curves are said to have a Contact of the SecondOrder. If (a) = xl> (a), 4/(a) = iff (a), (a) = «/>(», and also, Contact of Curves Envelopes 195 the curves are said to have a Contact of the Third Order; andso on. Hence, generally, if *(*)=*(«), *(«)=/(«), 4>(«)=*(*) • • • <F(<*)=r(«) ...(*) the curves have a contact <?/* /-fo nth order. Cor. i . A contact of the nth order involves n + i conditions. Cor. 2. As only ?i + i conditions can 01 general be imposedupon a locus whose equation contains n + i arbitrary constants,the highest order o
. Differential and integral calculus. e curves are said to have a Contact of the SecondOrder. If (a) = xl> (a), 4/(a) = iff (a), (a) = «/>(», and also, Contact of Curves Envelopes 195 the curves are said to have a Contact of the Third Order; andso on. Hence, generally, if *(*)=*(«), *(«)=/(«), 4>(«)=*(*) • • • <F(<*)=r(«) ...(*) the curves have a contact <?/* /-fo nth order. Cor. i . A contact of the nth order involves n + i conditions. Cor. 2. As only ?i + i conditions can 01 general be imposedupon a locus whose equation contains n + i arbitrary constants,the highest order of contact to such a curve is in general thenth. Thus the straight lines Ax -f By -+- C = o, having onlytwo arbitrary constants, can have a contact of the ist order; thecircle (x — df -f- (y — b)2 = r2 having three arbitrary constantscan have a contact of the second order. 142. Two curves in contact do or do not cross each other at theircommon point according as their order of contact is even or Let y = (x) and y = \p (x) be the equations of two curveshaving the nth order of contact, and let x = a be the abscissaof their common point B, Fig. 30. Let us add a small incre-ment h to a, then 4>(a + X)-$(a + h) is the difference between the ordinates of the two curves corre-sponding to the same abscissa x = a + h. Expanding both 196 Differential Calculus terms in the expression by Taylors Theorem, and collectinglike derivatives, we have (since cf> (a) p= if/ (a)) hi(a+h)-^(a+h) = \ 4>(a) - #(a) \h + \ (a) - i[/(a) ] * Li 5 + $+>)_f («)}* + (x) i. If n is even, then cf>f(a) = f (a), (a) = ip(a)t ... n (a) = f1 (a), and the terms in the second number of (i) successively vanishuntil the (n -f i)th term is reached. As this term contains haffected with an odd exponent (n ?+- i) its sign will change withA, and if h is taken sufficiently small the numerical value of thisterm will exceed the sum of all the other terms. Hence
Size: 2166px × 1154px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1918
