52 !/.>//' OMOMXTBY 



From what has just been said it is clear that y = if z = 0, IP-MO- th<> 

 *iirv goes through the mum; \\hen jc increases rout inimiisly from o to 

 *. y increases continuously from to GO, but when x increases (lir<i,/f, ". 

 y panes suddenly from + co to co, and tln> rum- is tli*c<mttnn<,>. 



value of x. So also when x increases ontimmnsly from ^ to '-, 

 y increases continuously from - oo through to 4 co, and is again dis- 

 continuous for x = '-^. The locus consists of an infinite number of 



infinite, but continuous branches, separated by the points of discon- 

 tinuity for which * =:j> x=^, ar=^, 



The otli-r trigonometric function^, y sin .r, ;/ = sec x, etc., can all be 

 plotted by a method analogous to that al-o\.-. 



EXERCISES 



Construct and discuss the loci of the following equations : 

 , *?_3?_i 3. y = secx. 7. r = sin it. 

 49" 4. ^-y^a*. 8. x'-!-y* = 0. 



-?^ ? -- ::;;;//=:: ^5^-* < 



38. The locus of an equation remains unchanged: (a) by 

 any transposition of the terms of the equation ; and ( p) by 

 multiplying both members of the equation by any finite con- 

 stant. 



(a) If in any equation the terms are transposed from nn<- 

 member to the other in any way whatever, the locus <i tin- 

 equation is not changed thereby ; for the coordinates of all 

 the points which satisfied the equation in its original form. 

 and only those coordinates, satisfy it after the transjio>itions 

 are made. [See Art. 35 (1).] 



() If both members of an equation are multiplied by any 

 finite constant &, its locus is not changed thereby. For if 

 the terms of the equation, after the multiplication has been 

 performed, are all transposed to the first member, that mem- 

 ber may be written as the product of the constant k and a 



