. An elementary treatise on the differential and integral calculus. Fig. 41. in which the variable parameters a and b are connected bythe equation a2 + #2 = c2. (2) Differentiating (1) and (2), regarding a and b as varia-ble, we have -0 da — Tkdb = 0, ora2 b2 -, da = ^ t&a2 W ada -f 5r?5 = 0, or — ada = (3) by (4), we have (3) x y x y5 _ y a b a b ^3— J8» OT ^ ~ T2 ~ «2 + ^ a = (rcc2)^ and J = 1 c*5 234 EXAMPLES. which in (2) gives ,\ d* + #t — cf^ which is the equation form of the locus is given inFig. 42, and is called a hypo-cycloid,which is a curve generated by ap
. An elementary treatise on the differential and integral calculus. Fig. 41. in which the variable parameters a and b are connected bythe equation a2 + #2 = c2. (2) Differentiating (1) and (2), regarding a and b as varia-ble, we have -0 da — Tkdb = 0, ora2 b2 -, da = ^ t&a2 W ada -f 5r?5 = 0, or — ada = (3) by (4), we have (3) x y x y5 _ y a b a b ^3— J8» OT ^ ~ T2 ~ «2 + ^ a = (rcc2)^ and J = 1 c*5 234 EXAMPLES. which in (2) gives ,\ d* + #t — cf^ which is the equation form of the locus is given inFig. 42, and is called a hypo-cycloid,which is a curve generated by apoint in the circumference of acircle as it rolls on the concavearc of a fixed Fig. 42. 3. Find the envelope of a series of ellipses whose axes arecoincident in direction, their product being constant. Here x2 , y2 _ i a% -*- 02 - x- (i) Let a-h = c; (2) • • x% j , y2 tl a x2 j y2 77,—a da + f-Q t/0 = 0, or -da= — f-Q J3 a3 J3 (3) ^ dh n da dh h -T- = 0, or — = — -j- • ah a h (4) Dividing (3) by (4), we have oft ifr -g = ^ = i, by substituting in (1). from (2), and h = ±3/a/2; ay = ± 2 which is the equation of an hyperbola referred to its asymp-totes as axes. This example may also be solved as follows: Eliminatingb from (1) and (2), we have EXAMPLES. 235 X2 i_ a% 2 1, (5) in which we have only the variable parameter #. x2 aif __ 9 cz y (6) which in (5) gives 2 + S = i- ••• »y = ¥- 4. Find the envelope of the right lines whose generalequation is y = mx + (ahri2 + 62)*, (1) where m is the variable parameter. h t We find m = a \/a2 — x* x^ XI2 which in (1) gives —a + j-2 = 1 for the required envelope. Hence the envelope of (1) is a
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