Elements of analysis as applied to the mechanics of engineering and machinery . ;, a diminution of ^, giA^es an increase of COS. X. If SP be drawn at rio-ht angles to C Il^A there results a triangle SP1\ which, on ac-count of the equality of the angles PTS and CQNoi (7Pif, issimilar to the triangle CP31^ and hence, we have:ST__ CP . 8 ta7ig. X __ 1 Yr ~ Inr ^ ^ ^ ~ COS. X. U M But we have also:III. d(tang. x) = secayit. xdx 1 c) X , hence SP = and COS. X COS. X (cos. xy- TV f 7Z If, for ^, we substitute — — ,37, therefore, for 8-^, 8 I -^r-— 8^, we obtain: 8 tang. { ~ X \ X ex [co,. (-^ - ^)]
Elements of analysis as applied to the mechanics of engineering and machinery . ;, a diminution of ^, giA^es an increase of COS. X. If SP be drawn at rio-ht angles to C Il^A there results a triangle SP1\ which, on ac-count of the equality of the angles PTS and CQNoi (7Pif, issimilar to the triangle CP31^ and hence, we have:ST__ CP . 8 ta7ig. X __ 1 Yr ~ Inr ^ ^ ^ ~ COS. X. U M But we have also:III. d(tang. x) = secayit. xdx 1 c) X , hence SP = and COS. X COS. X (cos. xy- TV f 7Z If, for ^, we substitute — — ,37, therefore, for 8-^, 8 I -^r-— 8^, we obtain: 8 tang. { ~ X \ X ex [co,. (-^ - ^)] 1. e. lY. 8 (cotang x) = — 8^ (sin. xYBy inversion, these formulae give, for the differential of the arc: 8^ = 8^ 8 sin. Xcos. X 8 COS. X sm. X = (cos. xy 8 tang, x 8 sin. X = — (sin. xy 8 cotang. x, or;8 tang, x V \ — (sin. xy 1 + (^^^^9- ^y 38 ELEMENTS OF ANALYSIS. [Art. 2G. , ^ 8 COS. X d cotanq, x as also, ox = — = — ~ -. 1/ 1 _ (^cos. xf 1 -h {cotang xf If now sin, x be designated by y, and x by arc. (sin. = ^/),we obtain: Y. 8 arc. (sin. = y) = ^^ ; and in the same manner we find: 8 VYI. 8 arc. (cos. = y) z= — -— ^ , as also: y 1 — ?/^ YIL 8 arc. (tana. = i/) rz= —, and V y J^ 1 + ^ 8 ?/YIII. 8 arc. (cotang. z= y) = 1 + y^ Art. 26. The last differential Ibrmnlae give, by inversion, thefollowing integral formnlae: I. J COS. X d X = sin.
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