Page:Encyclopædia Britannica, Ninth Edition, v. 15.djvu/695

Rh If the body receives both the rotation 77 and the rotation 77 then the joint effect of these two rotations must be equal to that of a single rotation &amp;lt;, so that dX dX

^Y = ^Y &amp;lt;2Y d&amp;lt;f&amp;gt; dr, dr, dZ dZ dZ d&amp;lt;p dr, dr, or uniting the two sets of equations we have dX Y-Lrt 7 -r- = a X + +f Z As this movement must also be a rotation, the three right-hand mem bers must be capable of being rendered zero for certain values of X, Y, Z, and therefore we have (remembering that o &quot;o = 0. 2 &amp;gt; . - This condition reduces to coa ( (/! &&amp;gt;) - or a co 2 +eo(a 1 i - a v 1 ) = 0. This equation must be satisfied for every value of u ; for, whatever be the amplitudes of the two rotations, they must when com pounded be equal to a single rotation. We therefore have the conditions &quot;i^e =. To satisfy the latter condition either o^ or v g must be equal to zero. We must examine which of these two conditions is required by the problem. Since o 8 is equal to zero we have dX dY dZ If t&amp;gt; were zero then the first equation would show X to be con stant ;-and the result would be that Y = 1^X77 + ^o-j^Xr; 2 -f const. ; or, in other words, Y would be susceptible of indefinite increase with the increase of 77. The supposition j/ = is therefore precluded, and we are forced to admit that o 1 = 0. The three equations then reduce to = Z =vZ dr, l dZ _ ,, If the body receives a rotation 77&quot; about an axis which leaves X and Y unaltered, we then have The condition that the two roots of h shall be purely imaginary gives us /o + fc-0. Let this rotation and the first rotation be communicated together. The resulting rotation could have been produced by a rotation x&amp;gt; and thus we have ^X = ^X + ^X dx dr) drj&quot; ^Y = ^Y + dY dx dr, dr,&quot; dZdZ dZ Substituting, we obtain as before and as before the condition must be fulfilled fo &amp;gt; 0o &amp;gt; A, Si or, expanding, This can only be satisfied for all values of if / = and if To determine whether g Q can be zero, we have the equations dX dY dr, It can be shown that if g were zero then we should have Z capable of indefinite increase ; and hence we see that/ 2 must be zero, so that the three equations have the form Let us now see whether these equations will fulfil the necessary condition for a rotation . If ^

dX drj&quot; = _ dty dr) dri&quot; dZ dZ dZ dty dr) dr)&quot; we shall then have by substitution But, if this is to represent a rotation, As this is always to be true, even suppose g, /,, and g&amp;lt; 2 for instance were multiplied by a common factor, it is plain that we must have and BS^O&quot;! = ^ The first condition is to be satisfied by making g 2 zero ; for neither of the other possible solutions is admissible if the coordi nates are to be presented from indefinite increase. In a similar way the second condition requires that v l must be zero. Resuming now the three groups of equations, they are as follows: ^7V ^7V .JV CtA. _ W-.A Cl-A. ._ df) dY 7 -r- = - UlL dr, dZ v -7- = +oiY dr, drf fr dZ v J- =2 X Finally let us suppose that the body receives all three motions simultaneously. The resulting motion must still be a rotation, and thus we have the condition , ff &amp;gt; v o /x, , -a. a.,. + u , = 0, 