18 ON COMPOUND PROJECTION IN TWO DIMENSIONS. 



tive axis and the tangent at any point to the primitive 



curve, we have 



, ,dx dv 

 cos0 = Z^+m^; 



therefore, by substitution, equation (lo) may be put in 

 the form 



VdS:^^ + dsy"- = i/ 1 + cos^'i^ . ds. 



If we suppose the primitive area to revolve round any 

 axis perpendicular to its plane, since the primitive axis is 

 rigidly connected with it, the expression \\/ 1 + cos^^ . ds 

 will be invariable ; replacing it by c, we shall have then 



j s^ ds^-\-dSy^ — c. 



Relation of Projectrices of Higher Orders. — From a 

 primitive may be derived two projectrices ; but each of 

 these may in its turn be regarded as a primitive that may 

 be operated upon in a similar manner ; then, on a repe- 

 tition of the process, we shall obtain four projectrices. 

 The relation of the area of these to that of the primitive 

 may be obtained as follows. A^j. being the primary pro- 

 jectrix on the axis of x, the secondary projectrices which 

 may be derived from it may be denoted by {k^^ ^^^d 

 (Aj;)y, and we shall have 



(A:,);, = m^K, (ii) 



{K)y = m'-l'-A (12) 



If (A^)a; and (A^),, denote the secondary projectrices 

 which may be derived from A^, we shall have 



(A,), = /WA, (13) 



(A,), = /^A (14) 



