24 VARIABLES AND FUNCTIONS. [iNT. II. 



AI, h 2 ...... of the functions q 1} g 2 ...... and the partial derivatives 



, ~ , ...... we lay off the partial parameters 



in the directions /^, h 2 ...... or their opposites, according as ^ >0, 



OQi 



or the opposite, and find the resultant of P 1} P 2 , ....... 



If the functions q lt q z , ...... are three in number, and form an 



orthogonal system, the equation 



?-?,+?,+? 



gives for the modulus, or numerical value of the parameter 



Examples. (1) in 14. Let the distance of M in the given direction 



/V ffy 



from the plane be u. AF= AM = - -, where a is the angle between 



cos a 



the given direction and that of a perpendicular to the given plane. 



_ 



Aw cos a ' cos a ' 



If the given direction is perpendicular to the given plane P=l. 

 Accordingly for q 1 = x, q 2 = y, q 3 = z, the rectangular coordinates of a 

 point, we have P x = P y = P z = lj and for any f unction f(x, y, z) 



The projections of P on the coordinate axes are the partial parameters 



Consequently, if cos (sec), cos (sy), cos (52;) are the direction cosines 

 of a direction 5, the derivative in that direction 



= P 1 cos (sec) + P 2 cos (sy) -f P 3 cos (c) 



ar ar . x ar 



= cos (sec) + -T- cos (8y) + -T- cos (*). 



003 07/ 02 



P may be written in terms of the unit vectors i, j, k as 



