The rare lichens of Glencroe / by Hugh MacMillan.

  • Macmillan, Hugh, 1833-1903.
Date:
[between 1890 and 1899?]
Catalogue details

Licence: Public Domain Mark

Credit: The rare lichens of Glencroe / by Hugh MacMillan. Source: Wellcome Collection.

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    other hand, the theorem will not be deduced so entirely from: first principles. From (2) it follows immediately that — f dv—l TT J V if x <a] = 0 if x = a\ and = — 1, if x> a, and therefore 7T r sin (a + 7* — x) v sin [a — x)v Jo V v dv = 1, if x > a and < a + 7* = if £C = a, or = a + 7* = 0, if x<a, or > c? + 7*, the expression in square brackets under the integral sign is, when 7*, is made infinitesimal, = 7* cos (a — x)v] multiply then both sides by cf (a), which, if x lies between a and a-f-7*, is the same as cf> (x), and we have 1 f00 — | cos(a - x) vdv.Jicf) (a) = cf> (x), if x> a and < a + 7* — ^cf> (cc), if x = a, or — a + A = 0, if x < a, or > a + 7*.. Similarly 1 f00 — I cos (a + 7* — x) vdv.hcf) (a + A) = «/) (a?),. 7r •'o cc > a + A and < a 4- 2A, &c., 1 f00 — I cos {a + 27* — a;) vdv.Jicf) (a + 27*) = <f>xr 7r J0 a; > a -f 27* and < a + 37*, &c., &c. &c. Now the value of 7* {cos (a — cc) vcf> (a) + cos (a + 7* — a?) vcf> (a + 7*)... + cos-(a + nh — x) vcf> (a + nh)}r is I cos (a — cc) v<f> (a) da^ (taking a =7?, a + nh = q) and we Ip therefore have by addition — f I cos (a — x) vcf> (a) da dv = cf> (x)j if a; and < <7 7I’ J P J 0 1 * / \ • n = (x), if x =y», or = q — 0, if x <y>, or > q (3). if if