for determining Frequencies of Lateral Vibration. 425 



specified section, and \ being a non-dimensional coefficient. 

 From this point of view, our problem is to determine pos- 

 sible values for X, and we are free to change our scale of 

 length in any convenient manner, whilst the results will be 

 applicable to a whole family of rods, differing in actual 

 length, and specified simply by the variation of the ratios 

 -3/^o an d pIpQ with the ratio xfl : the natural frequencies for 

 different members of the family will vary inversely as the 

 square of the length of the rod. 



In the light of these remarks we may now proceed to de- 

 velop our construction. In fig. 1. (Pi. VII.) we choose axes of 

 Y and x. as shown, and take OA to represent /, the length of 

 the rod (or the distance from to the next section of con- 

 straint, if the rod is continuous); and we draw vertical lines 

 to divide OA at the points 1, 2, 3, ... . into any convenient 

 number, ?i, of equal parts (;i=10 in the diagram shown). 

 The values of 23 are assumed to be known at the points of 

 subdivision, and lengths b ]? b 2 , .... are taken, as shown in 

 fig. 2, proportional to ®i, 2>2> • • • • e ^c. Similarly, in fig. 3 

 we choose axes O'ni and O'x for a curve of which the ordi- 

 nates HI are to be made proportional to M, and continue the 

 vertical subdividing- lines ; and in fig. 4 we take lengths 

 k 1? k 2 , .... etc. proportional to the values of K which corre- 

 spond to the points of subdivision. It should be noted that 

 the actual scale on which HI represents M is for the moment 

 left undetermined. 



The construction then proceeds as follows : — Oa and Q'a' 



dY „ , dM 

 dx 



are drawn to represent the values of Y, -j— , M, and 



ax 



at the end x~0 (figs. 1-4, PL VII. are drawn for an example 

 in which it is specified that both Y and M vanish at this end : 

 the slopes of Oa and OV are for the moment immaterial) ; 

 and the polar diagrams (figs. 2 and 4) are begun by drawing 

 pOj parallel to Oa and p'Oi parallel to O'a', the points Oi 

 and 0/ being distant by amounts b l and k 2 respectively 

 from the verticals through p and p' . We then make pq in 

 fig. 2 equal to a'V in fig. 3, and p q in fig. 4 equal to a\ 

 in fig. 1 ; join q0\ and ^'0/, producing them to 2 and 2 ' 

 as shown, and continue figs. 1 and 3 by drawing ah and a'b' 

 parallel to q0\ and q'Oi respectively*. 



The processes just described are repeated at every section 

 of subdivision : viz., qr is made equal to b'2' and q'r' to b'2, 

 be and b f c' are drawn parallel to 2 ?' and 2 V respectively, 



* Dashes have been accidentally omitted from the letters/ in fig, 3 

 And i; O2,. . . etc. in tig-. 4. 



