AOT) DETEEMINiJfTS OE LINES. 
475 
But since the lines _L' to Ox) and {'mq^ to Oy) are respectively proportional and 
perpendicular to and Q^, and since the latter lines have Q for their complete sum, it 
evidently follows that the former two lines have for their complete sum a line which is 
perpendicular to Q, and whose length is rsq. 
Hence 
D^(Q)= II to II toO^) +(=^2 _L' to Q) (IL) 
17. This last formula is true whether the axes be rectangular or oblique, and may be 
made the basis of all the formulae of relative motion in one plane. 
It may be observed that the line 
(%||toOif) + (^||toOj,) 
is what would be the complete differential coefficient of Q if the coordinate axes were 
fixed ; and it may therefore be called the complete differential coefficient relative to the 
moving axes, or, more briefly, the relative differential coefficient of Q. So that the above 
formula shows that the comiglete differential coefficient of Q is its relative differential 
coefficient together with a line {ysq _1_^ to Q), the latter line being drawn towards the direc- 
tion in which the axes are revolving. 
18. If the axes of coordinates be rectangular, then the line _L'‘to O^) is evidently 
the same as f^q^ || to Oy \ and the line {mqy to Oy) is the same as {—•^qy 1| to O^) ; and 
therefore, looking at the equations (I.) in section 16, we see that 
D,(Q)=D,(QJ+D,(Q,) 
In other words, the components of Di(Q) parallel to Oa? and Oy are respectively 
“d §+=’2. (III.) 
The same result may be also deduced from formula (II.) in the same section, if we 
resolve the line (^-g' to Q) along the rectangular axes of x and y. 
19. Let us apply the above formulae first to the velocity of a particle. 
Suppose, then, a particle to move in a given plane, and that the rectangular axes of 
coordinates in that plane revolve about the origin with an angular velocity m at time t. 
Let v^ and Vy be the components of the particle’s velocity along the moving axes. Then, 
since the velocity is the complete differential coefficient of the radius vector, and since 
X and y are the components of that radius vector, it follows from the formulse (III.) of 
the last section, that 
dx 
I 
