We report experimental observations of an undulational instability of myelin figures. Motivated by this, we examine theoretically the deformation and possible instability of concentric, cylindrical, multi-lamellar membrane structures. Under conditions of osmotic stress (swelling or dehydration), we find a stable, deformed state in which the layer deformation is given by δR ∝ r √ B A /(hB) , where BA is the area compression modulus, B is the inter-layer compression modulus, and h is the repeat distance of layers. Also, above a finite threshold of dehydration (or osmotic stress), we find that the system becomes unstable to undulations, first with a characteristic wavelength of order √ ξd0, where ξ is the standard smectic penetration depth and d0 is the thickness of dehydrated region.PACS numbers: PACS numbers: 87.22. Bt, 68.15.+e The swelling of bulk phospholipid by water at a temperature higher than the chain melting temperature can lead to the growth of multilayer lipid tubules from the interface of bulk lipid and water [1,2,3,4,5]. These tubules are of mesoscopic dimensions with a diameter about 20-40 µm. They have the symmetry of a smectic-A liquid crystal and a strong optical anisotropy. Both electron microscopy [6] and X-ray diffraction show a concentric multilayer structure of these lipid tubes with a layer spacing about 60Å [7]. Due to the similarity of their structural features to those of nerve myelin sheaths, these tubules are often referred as "myelin tubes" or "myelin figures". The morphological features of myelin tubes have been classified into two steps according to the time of growth [3]. During the first step, simple tubes grow into bulk water with an initial growth rate in length about 1 µm/sec. The diameter and number of layers of tubes appear to remain constant during this process and the growth rate is inversely proportional to the square root of growth time [2,4]. The growth almost stops in the second step which is characterized by the formation of complicated morphologies of myelin tubes, such as helical and coiling forms, apparently in order to maximize their inter-membrane attraction [8].In our experiments, we have observed an instability of myelin tubes under dehydration by isolating a single myelin tube from others (during the first step, as shown in Fig. 1) and allowing the bulk water to evaporate in a controlled open chamber. During this dehydration step, periodic bumps with a wavelength about 1 µm are observed on the surface of myelin tubes as shown in Fig. 2(a). As water further evaporates, these bumps grow into arms, as shown in Figs. 2(b) and 2(c), and a similar instability can occur for these long arms. As far as we know, this is the first reported observation of the instability of myelin tubes under dehydration. After the bulk water dries out, the tubular structure of myelins disappear and a mosaic structure is observed. Although the full structural change of myelin tubes under dehydration is very complicated and highly nonlinear, the initial instability of myelin tubes under...