March 17, 1899.] 



SCIENGE. 



393 



right and down, down and left, left and up, 

 while the particles at the origin run through 

 all phases together. This case corresponde 

 to the preceding for 270°, etc. 



All this is evident enough ; but it is, 

 nevertheless, advisable to make a diagram 

 of the position of the pointers as here 

 shown, in order instantly to discern the 

 phases iu which the initial particles meet in 

 any case. In the table the positions of the 

 pointers are designated by arrows ; + in de- 

 notes maximum displacement, etc. Further 

 explanation will be given presently. 



11. Space Waves. — The composition of two 

 simple harmonics at right angles to each 

 other will necessarily require special treat- 

 ment, for here the rear riders are at right 

 angles to those of the former case, and the 

 S. H. motion of the rear axle is not reversed 

 at the balls. If displacement up and for- 

 ward from the observer's view be consid- 

 ered positive, then the null position or zero 

 of phase of the particles at the origin corre- 

 sponds to pointers left for the front axle and 

 up for the rear axle, as seen by the operator. 

 The compound simple harmonic of these 

 components is thus a linear vibration with 

 amplitude ^2, as regards the equal com- 

 ponents, and making an angle of 45° to the 

 horizontal from the observer to the machine. 

 It thus lies in the first quadrant, as seen by 

 the operator at the crank. 



Both component 8. H. curves leave the 

 origin with a descending node. 



Hence, if in the above table we shove the 

 first column of entries one row ahead, i. e., 

 if we begin for no phase difference with the 

 second row and continue in cyclical order, 

 the table will be adapted to the present case. 

 Pointers in opposite directions will thus 

 correspond to counter-clockwise circular 

 motion in the compound wave ; pointers in 

 the same direction to clockwise circular 

 motion, as seen by the operator at the crank. 

 The first of these cases will, however, cor- 

 respond to a right-handed, the second to a 



left-handed, screw when seen from the ori- 

 gin, since all waves move from left to right. 



The table contains an entry relative to 

 the present case. It thus indicates 16 car- 

 dinal phase differences for plane and the 

 same number for space waves. 



12. Effective Circles of Reference. — -Finally^ 

 a word may be said as to the position of the 

 circles of reference corresponding to the two- 

 component S. H. motions. Clearly, the cen- 

 ters of the eccentrics (marked in Fig. 1) de- 

 termine the amplitude of the 8. H. M. In 

 all phases, however, the riders are nearly 

 normally above or else to the rear of these 

 centers by a distance equal to the radius of 

 the eccentric, and, therefore, always in the 

 same kind of reciprocating motion which 

 corresponds to the amplitude and period of 

 the eccentric. 



Hence the circle of refei'ence of the ver- 

 tical 8. H. M. is on a vertical diameter and 

 tangent to the highest and lowest positions 

 of the edge of the eccentric on the same side 

 of the axle. The diameter prolonged passes 

 vertically through the cam axis, and its 

 length is twice the throw of the center of 

 eccentric. This circle of reference for the 

 horizontal 8. H. M. of the riders (displace- 

 ment + rearward) is on a horizontal diam- 

 eter and tangent to the extreme right and 

 left positions of the edge of the eccentric on 

 the same side of the axle. 



The amplitude of the vertical vibrations 

 is modified by the lengths given to the ex- 

 tensible levers. If I be the lever length 

 between the axles and I' that beyond the 

 axles, and if a, a' denote the front and 

 rear amplitude at the eccentrics, then the 

 effective amplitude at the particles will be 

 a(l + V) jl and a'l' /I, and their ratio 



a' I' 

 a I+T 



may be varied at pleasure from zero to 

 about 9/8, since I' is the extensible part. 

 Usually the ratio one is desirable. 



