ANALYTIC gXOMKTltY [Cii. V I. 



EXERCISES 



1. r.ivon the equation Or*- 16y* = 144 referred to rectangular axes; 

 what does this equation becom*- if tnuiHfonned to new axes such that the 

 new x-axis makes the angle tan- 1 ( - |), and the new y-axis the angle 

 tan~ l (f), with the old z-axis, origin unchanged? 



2. If the old and new ar-axes coincide, and the new axes are rectan- 

 gular while the old axes are inclined at an angle of 60, what are the 

 equations of transformation from the old axes to the new ? From the 

 new axes to the old ? Origin unchanged in each case. 



3. If the first two of the three sides of a triangle whose equations are 

 2y + z + l=0, 3y-x-l=0, and 2 x -f 3 y = 1, are chosen as new axes, 

 find the new equations of the sides. 



74. Transformation from one set of oblique axes to another, 



origin unchanged. Let OX 

 and OY be a given pair of 

 axes, OX' and OY' the n NV 

 axes, and let the angles XO F", 

 X'OY', XOX', and XOY' 

 be denoted by ft), a/, 0, and <, 

 respectively. Also let P. ; 1 1 1 y 

 point in the plane, have the 



coordinates x and y when referred to the first pair of axes, 

 and x 1 and y 1 when referred to the second pair. 



Draw M 1 P parallel to 0F', MP and QM' para IK -1 to 

 OY, and M'R parallel to OX. 

 Then, from the triangle OQM 1 , 



OQ = x' 8[n ( < *- 0> > and QM'=x*?^, 

 sin CD sin CD 



and from the triangle RM'P, 



RM< = y* 8ip <y-) and RP = 1 



81110) ' S'llft) 



But OM = OQ - RM\ and MP = QM 1 + RP ; 



