A Treatise on the Theory of Screws

FREEDOM OF THE FOURTH ORDER. 231 The co-ordinates of 0 are proportional to a"'«6 + £%86 + 7"^6 + 3'"86. As 0 is to be a principal screw of inertia it follows that the expressions last written multiplied severally by plt ..., p* must be proportional to the intensities of the impulsive wrenches received by the body : whence we have the following equations in which h is a quantity which is the same for each of the co-ordinates. hpl (a"'«! + + 7wyi + S"^) = a "a, + ß'"ßl + y'"y1 + 8"'81 + X"^ + p'"^, hp6 (a"'as + ß'"ß6 + y"'ye + 8'"8e) = a"ae + ß”'ß6 + y'"ye + 8"'8e + X'"X8 + p'"ps. We are now to multiply these equations by au respectively, and add. If we repeat the process using ßlt ß6; yi, y8; 86; X1( ...,X6; plt...,ps and if we remember that a is reciprocal to ß, y, 8 because the system is co-reciprocal and that a is reciprocal to X and p because X and p belong to the reciprocal system, then observing that like conditions hold for ß, y, and 8, we have the equations (Xa13-/(:ptt)+iS"'SaA +7"'^aj7i + 6"'2ai8i +X'"Xa1X1 +p"'Xa1p1 =0, SaA +ß'"(tß1*-hpll) + y"'2ß1y1 + 8/"W1 +X'" XßiXi+p" Xß^p^ = 0, SaiYi +ß"'Sy1ß1 +yW~ hpj+8"'ty& +xz/ P' "Syi/ij =0, SaÄ +ß','X81ß1 +7'"S817i +8''\t8^-hps)+X,"t81X1 +/z."'S81^1 =0, SaAj +ß','Xx1ß1 +y"'ZX1y1 + 8,"X\81 +x""£x? +p =0, +ß'"'£p1ß1 +y"'lp1y1 + 8'"lpA ■+ x'" Xp1x1+p' "Xpi =0. From these equations a'", ß"', y'", 8"', X"', p" can be eliminated and the result is to give a biquadratic for h. Thus we have the four roots for the equation. Each of these roots will give a corresponding set of values for a"'; ß"'t ry'", S'", X'", p"; thus we obtain a"'ai + ß"'ßi + 7/z/Vi + ^i> a"'a6 + ß"'ß<s + y'+ 3"'é6, which are proportional to the co-ordinates of the corresponding principal screw of inertia. The values of X'" and p" determine the impulsive reaction of the con- straints. 221. Application of Euler’s Theorem. It may be of interest to show how the co-ordinates of the instantaneous