GRAPHICAL ALGEBRA. 93 



with the greatest ease find all data regarding direction and intensity of the vector- 

 sum in all points of the two curves (a) and (b). 



The curves (a) and (b) belong to the first set of auxiliary curves drawn for the de- 

 termination of the vector-sum, section 158 (i). While we draw the curve (a) we can 

 mark on it the points where it will be cut by the required curves <p = o, 1, 2, . . . 

 for these will be the same points as those which serve for the determination of the 

 curve (8 a = o itself, namely, the points of intersection of the curve a = o with /3 = o, 

 of the curve ai with /3 = 1, of the curve a = 2 with /J = 2, and so on (fig. 77 c). 

 In order to find the points where the same curve is cut by the required intensity- 

 curves F = o, 1, 2, . . ., we have simply to draw the short parts of the curves 

 A-\-B = o, 1, 2, . . . which cut the curve (a) (see fig. 77 d). 



In the same manner, while we draw the curve (b) , we can mark on it the points 

 where it will be cut by the curves <p = 0, 1, 2, . . . ; for these points will again be 

 the same as those points of intersection of the given curves a = const, and /3 = const, 

 which serve to determine the curve (b). We have to remark that the integer values 

 of <p which should be noted at these points will be those of a when A>B and 

 those of (8 when B>A. In order to find the points where the same curve is cut by 

 the intensity-curves F=o, 1, 2, . . . , we have to draw the parts of the curves 

 \AB\ = o,i,2, . . . , which cut the curve (b) (see fig. 77 d). Evidently the inter- 

 section of the curve A B = o with the curve (b) gives the singular points of the field 

 of the vector-sum. It should be observed that the curve B A = o is identical 



with the curve -j = 1 , which we have used already in the preceding section for the 



determination of the singular points. These points will divide the curve (b) into 

 distinct branches. As we pass a singular point the value of the angle <p will change 

 suddenly from <p = a to tp = fi, or vice versa. 



Thus the investigation of the two curves (a) and (b) gives with great ease both 

 the situation of the singular points in the field of the vector-sum and in addition a 

 great number of points through which different curves representing the field of the 

 vector-sum shall pass. These data can be utilized in different ways, according 

 to the method otherwise used for finding the field of the vector-sum. If the method 

 given in the preceding section be retained, it will be important to remark that the 

 curves (a) and (b) will turn up again as the curves 



(c) <p a = o 



(d) <p-a = 32 



in the auxiliary field of the angle <p a, which is found by the operation (3) of the 

 preceding section. The curve (c) will correspond to the curve (a) and certain parts 

 of the curve {b), the change in the correspondence taking place at the singular points. 



160. Graphical Tables for Vector-Addition. Our first solution of the problem 

 of graphical vector-addition depended upon the decomposition of the general 

 problem with four variables into six partial problems each with two variables. 

 But if we use the method which we have developed in section 152 for three varia- 



