GEOMETRICAL REPRESENTATION OE GROWTH. 33 



cells which are not exhibiting any phenomena of growth. The sur- 

 face tension is sufficiently great to warrant the neglect of the weight 

 of the mass of the spore, which would tend to destroy the spherical 

 form if the densities were slightly unequal ; thus so far as observa- 

 tions can go, the spores are absolutely spherical ; but no proof of 

 this exists unless the mechanical theory of surface tension can be 

 applied. That is to say, the absolute proof of the shape assumed can 

 only he determined iy physical deductions and not simply hy observation. 



In the same way, no amount of actual measurement of a specimen 

 would convince a mathematician that the apparently parabolic 

 curves seen in sections were of the strict {y'^ = 4 ax) type, unless 

 some mechanical determining cause can be adduced in support of 

 such a statement ; as, for example, a hypothesis that the cells might 

 be regarded as homologous with projectiles discharged from the 

 growing apex. 



The paraboloid theory of Sachs stiU remains a good working 

 hypothesis, and will stand or fall as the theories based on it can or 

 cannot explain other allied phenomena ; its value depends on the 

 extent to which other facts can be deduced from it. 



Thus, if the section of the growing apex is a true parabola, over 

 which the superficial cells may be supposed to glide until they reach 

 a position of rest on the cylindrical surface of the full-grown stem, 

 it is possible that the motion in the particles composing the fluid 

 mass of protoplasm might be resolved into a transverse velocity and 

 a longitudinal acceleration ; the former, a steady uniform movement 

 due apparently to the expansion of growth ; the latter, the expression 

 of the constant action of some retarding force acting along the axis 

 of the paraboloid apex. 



In a simple case in which none of the particles were discharged 

 above the horizontal line, it is clear that a paraboloid of revolution 

 would mark out the enclosing curve of the line of fall of all of 

 them ; while if the particles are regarded as being discharged in all 

 directions, as in the faU of particles of a bursting shell, the 

 enveloping curve would still be such a paraboloid, so that it is 

 immaterial whether the initial point be regarded as situated on the 

 surface of the apex, or at the focus of the parabola, so far as the 

 main outline of the curve is alone concerned. 



c 



