and Loci of AjwUoniiis, ^c. 39 



magnitude comprelicnded between tliem^ or equal to one of 

 them^ or not comprehended between them, so accordingly will 

 the circle AOH corresponding cut jMjNI in two imaginary 

 points, in two real and coincident points, or two real and dis- 

 tinct points C. 



Porismatic Relations of Data. 



1. It is evident the problem becomes indeterminate 

 when the circle OAH becomes indeterminate, &c. Now, if the 

 circle PQG passes through A, and that the point O coincides 

 with A, then H on AL will also coincide with A; if, AL 

 does not coincide with OP ; and, as the chord OH on OP is 

 equal zero, the circle AHO must be infinitely small. But if 

 AL and OP coincide, then, although AH and OH are 

 " infinitely smalls,^^ they lie both on AL, and therefore it is 

 evident that any circle touching AL in A is an answerable 

 circle AHO. Therefore in this last state of the data the 

 points C are innumerable, and the problem is '' porismatic," 

 as well as if we conceived O to move to A having OP parallel 

 AL, and thus causing H to be indeterminate when O coin- 

 cides with xV. 



2. If R coincides with S, and that we suppose Q, to approach 

 P until it comes to coincide with it, then G is at infinity, and 

 the straight line MIM lies in the infinite circumference PQ,G. 

 And for all values of j other than -f the point O coincides 

 with PQ, and one point C is coincident with PQ, and the 

 other point C with the point in which AL cuts MM. But 

 when ^ = [, then as the point O, and the point C coincident 

 with O, may be anywhere in M^I, the problem is said to be 

 " porismatic." 



Remarks concerning Particular Cases. 



1. Since PC.SD : QC.RD :: I : k, or, which is the same 

 thing, since 



PC.(R13 - RS) : (PC - Pa).RD :: I : k, therefore, 



when / = k, we have 



PC.RS = PQ.RD 



or PC : RD : : PQ RS 



Hence we derive a method of solving the problem — " Given 



the points P, R in the given straight lines MiM, NN; through 



two given points A and B to draw AI and BI making the 



angle I A right to B of a given magnitude right, and such that 



